diff --git a/doc/source/reference/extensions.rst b/doc/source/reference/extensions.rst
index bff5b2b70b518..eb2529716b55e 100644
--- a/doc/source/reference/extensions.rst
+++ b/doc/source/reference/extensions.rst
@@ -54,6 +54,7 @@ objects.
       api.extensions.ExtensionArray.interpolate
       api.extensions.ExtensionArray.isin
       api.extensions.ExtensionArray.isna
+      api.extensions.ExtensionArray.pad_or_backfill
       api.extensions.ExtensionArray.ravel
       api.extensions.ExtensionArray.repeat
       api.extensions.ExtensionArray.searchsorted
diff --git a/doc/source/whatsnew/v2.1.0.rst b/doc/source/whatsnew/v2.1.0.rst
index 0fdec3175f635..70cb0c0223eee 100644
--- a/doc/source/whatsnew/v2.1.0.rst
+++ b/doc/source/whatsnew/v2.1.0.rst
@@ -417,6 +417,7 @@ Other Deprecations
 - Deprecated positional indexing on :class:`Series` with :meth:`Series.__getitem__` and :meth:`Series.__setitem__`, in a future version ``ser[item]`` will *always* interpret ``item`` as a label, not a position (:issue:`50617`)
 - Deprecated replacing builtin and NumPy functions in ``.agg``, ``.apply``, and ``.transform``; use the corresponding string alias (e.g. ``"sum"`` for ``sum`` or ``np.sum``) instead (:issue:`53425`)
 - Deprecated strings ``T``, ``t``, ``L`` and ``l`` denoting units in :func:`to_timedelta` (:issue:`52536`)
+- Deprecated the "method" and "limit" keywords in ``ExtensionArray.fillna``, implement and use ``pad_or_backfill`` instead (:issue:`53621`)
 - Deprecated the "method" and "limit" keywords on :meth:`Series.fillna`, :meth:`DataFrame.fillna`, :meth:`SeriesGroupBy.fillna`, :meth:`DataFrameGroupBy.fillna`, and :meth:`Resampler.fillna`, use ``obj.bfill()`` or ``obj.ffill()`` instead (:issue:`53394`)
 - Deprecated the ``method`` and ``limit`` keywords in :meth:`DataFrame.replace` and :meth:`Series.replace` (:issue:`33302`)
 - Deprecated the use of non-supported datetime64 and timedelta64 resolutions with :func:`pandas.array`. Supported resolutions are: "s", "ms", "us", "ns" resolutions (:issue:`53058`)
diff --git a/pandas/core/arrays/_mixins.py b/pandas/core/arrays/_mixins.py
index 365f85a908099..359e0161e763c 100644
--- a/pandas/core/arrays/_mixins.py
+++ b/pandas/core/arrays/_mixins.py
@@ -18,6 +18,7 @@
     AxisInt,
     Dtype,
     F,
+    FillnaOptions,
     PositionalIndexer2D,
     PositionalIndexerTuple,
     ScalarIndexer,
@@ -295,6 +296,37 @@ def _fill_mask_inplace(
         func = missing.get_fill_func(method, ndim=self.ndim)
         func(self._ndarray.T, limit=limit, mask=mask.T)
 
+    def pad_or_backfill(
+        self,
+        *,
+        method: FillnaOptions,
+        limit: int | None = None,
+        limit_area: Literal["inside", "outside"] | None = None,
+        copy: bool = True,
+    ) -> Self:
+        mask = self.isna()
+        if mask.any():
+            # (for now) when self.ndim == 2, we assume axis=0
+            func = missing.get_fill_func(method, ndim=self.ndim)
+
+            npvalues = self._ndarray.T
+            if copy:
+                npvalues = npvalues.copy()
+            func(npvalues, limit=limit, mask=mask.T)
+            npvalues = npvalues.T
+
+            if copy:
+                new_values = self._from_backing_data(npvalues)
+            else:
+                new_values = self
+
+        else:
+            if copy:
+                new_values = self.copy()
+            else:
+                new_values = self
+        return new_values
+
     @doc(ExtensionArray.fillna)
     def fillna(
         self, value=None, method=None, limit: int | None = None, copy: bool = True
@@ -312,7 +344,6 @@ def fillna(
 
         if mask.any():
             if method is not None:
-                # TODO: check value is None
                 # (for now) when self.ndim == 2, we assume axis=0
                 func = missing.get_fill_func(method, ndim=self.ndim)
                 npvalues = self._ndarray.T
diff --git a/pandas/core/arrays/arrow/array.py b/pandas/core/arrays/arrow/array.py
index 7976f97cb49aa..88695f11fba59 100644
--- a/pandas/core/arrays/arrow/array.py
+++ b/pandas/core/arrays/arrow/array.py
@@ -899,6 +899,24 @@ def dropna(self) -> Self:
         """
         return type(self)(pc.drop_null(self._pa_array))
 
+    def pad_or_backfill(
+        self,
+        *,
+        method: FillnaOptions,
+        limit: int | None = None,
+        limit_area: Literal["inside", "outside"] | None = None,
+        copy: bool = True,
+    ) -> Self:
+        if not self._hasna:
+            # TODO(CoW): Not necessary anymore when CoW is the default
+            return self.copy()
+
+        # TODO(3.0): after EA.fillna 'method' deprecation is enforced, we can remove
+        #  this method entirely.
+        return super().pad_or_backfill(
+            method=method, limit=limit, limit_area=limit_area, copy=copy
+        )
+
     @doc(ExtensionArray.fillna)
     def fillna(
         self,
diff --git a/pandas/core/arrays/base.py b/pandas/core/arrays/base.py
index f69ddac4db6bf..325edba670fce 100644
--- a/pandas/core/arrays/base.py
+++ b/pandas/core/arrays/base.py
@@ -18,6 +18,7 @@
     cast,
     overload,
 )
+import warnings
 
 import numpy as np
 
@@ -33,6 +34,7 @@
     Substitution,
     cache_readonly,
 )
+from pandas.util._exceptions import find_stack_level
 from pandas.util._validators import (
     validate_bool_kwarg,
     validate_fillna_kwargs,
@@ -130,6 +132,7 @@ class ExtensionArray:
     interpolate
     isin
     isna
+    pad_or_backfill
     ravel
     repeat
     searchsorted
@@ -180,6 +183,7 @@ class ExtensionArray:
     methods:
 
     * fillna
+    * pad_or_backfill
     * dropna
     * unique
     * factorize / _values_for_factorize
@@ -907,6 +911,93 @@ def interpolate(
             f"{type(self).__name__} does not implement interpolate"
         )
 
+    def pad_or_backfill(
+        self,
+        *,
+        method: FillnaOptions,
+        limit: int | None = None,
+        limit_area: Literal["inside", "outside"] | None = None,
+        copy: bool = True,
+    ) -> Self:
+        """
+        Pad or backfill values, used by Series/DataFrame ffill and bfill.
+
+        Parameters
+        ----------
+        method : {'backfill', 'bfill', 'pad', 'ffill'}
+            Method to use for filling holes in reindexed Series:
+
+            * pad / ffill: propagate last valid observation forward to next valid.
+            * backfill / bfill: use NEXT valid observation to fill gap.
+
+        limit : int, default None
+            This is the maximum number of consecutive
+            NaN values to forward/backward fill. In other words, if there is
+            a gap with more than this number of consecutive NaNs, it will only
+            be partially filled. If method is not specified, this is the
+            maximum number of entries along the entire axis where NaNs will be
+            filled.
+
+        copy : bool, default True
+            Whether to make a copy of the data before filling. If False, then
+            the original should be modified and no new memory should be allocated.
+            For ExtensionArray subclasses that cannot do this, it is at the
+            author's discretion whether to ignore "copy=False" or to raise.
+            The base class implementation ignores the keyword if any NAs are
+            present.
+
+        Returns
+        -------
+        Same type as self
+
+        Examples
+        --------
+        >>> arr = pd.array([np.nan, np.nan, 2, 3, np.nan, np.nan])
+        >>> arr.pad_or_backfill(method="backfill", limit=1)
+        <IntegerArray>
+        [<NA>, 2, 2, 3, <NA>, <NA>]
+        Length: 6, dtype: Int64
+        """
+
+        # If a 3rd-party EA has implemented this functionality in fillna,
+        #  we warn that they need to implement pad_or_backfill instead.
+        if (
+            type(self).fillna is not ExtensionArray.fillna
+            and type(self).pad_or_backfill is ExtensionArray.pad_or_backfill
+        ):
+            # Check for pad_or_backfill here allows us to call
+            #  super().pad_or_backfill without getting this warning
+            warnings.warn(
+                "ExtensionArray.fillna 'method' keyword is deprecated. "
+                "In a future version. arr.pad_or_backfill will be called "
+                "instead. 3rd-party ExtensionArray authors need to implement "
+                "pad_or_backfill.",
+                FutureWarning,
+                stacklevel=find_stack_level(),
+            )
+            return self.fillna(method=method, limit=limit)
+
+        mask = self.isna()
+
+        if mask.any():
+            # NB: the base class does not respect the "copy" keyword
+            meth = missing.clean_fill_method(method)
+
+            npmask = np.asarray(mask)
+            if meth == "pad":
+                indexer = libalgos.get_fill_indexer(npmask, limit=limit)
+                return self.take(indexer, allow_fill=True)
+            else:
+                # i.e. meth == "backfill"
+                indexer = libalgos.get_fill_indexer(npmask[::-1], limit=limit)[::-1]
+                return self[::-1].take(indexer, allow_fill=True)
+
+        else:
+            if not copy:
+                return self
+            new_values = self.copy()
+        return new_values
+
     def fillna(
         self,
         value: object | ArrayLike | None = None,
@@ -921,7 +1012,7 @@ def fillna(
         ----------
         value : scalar, array-like
             If a scalar value is passed it is used to fill all missing values.
-            Alternatively, an array-like 'value' can be given. It's expected
+            Alternatively, an array-like "value" can be given. It's expected
             that the array-like have the same length as 'self'.
         method : {'backfill', 'bfill', 'pad', 'ffill', None}, default None
             Method to use for filling holes in reindexed Series:
@@ -929,6 +1020,8 @@ def fillna(
             * pad / ffill: propagate last valid observation forward to next valid.
             * backfill / bfill: use NEXT valid observation to fill gap.
 
+            .. deprecated:: 2.1.0
+
         limit : int, default None
             If method is specified, this is the maximum number of consecutive
             NaN values to forward/backward fill. In other words, if there is
@@ -937,6 +1030,8 @@ def fillna(
             maximum number of entries along the entire axis where NaNs will be
             filled.
 
+            .. deprecated:: 2.1.0
+
         copy : bool, default True
             Whether to make a copy of the data before filling. If False, then
             the original should be modified and no new memory should be allocated.
@@ -958,6 +1053,15 @@ def fillna(
         [0, 0, 2, 3, 0, 0]
         Length: 6, dtype: Int64
         """
+        if method is not None:
+            warnings.warn(
+                f"The 'method' keyword in {type(self).__name__}.fillna is "
+                "deprecated and will be removed in a future version. "
+                "Use pad_or_backfill instead.",
+                FutureWarning,
+                stacklevel=find_stack_level(),
+            )
+
         value, method = validate_fillna_kwargs(value, method)
 
         mask = self.isna()
diff --git a/pandas/core/arrays/interval.py b/pandas/core/arrays/interval.py
index 01f631b54a1d7..3263dd73fe4dc 100644
--- a/pandas/core/arrays/interval.py
+++ b/pandas/core/arrays/interval.py
@@ -29,6 +29,7 @@
     ArrayLike,
     AxisInt,
     Dtype,
+    FillnaOptions,
     IntervalClosedType,
     NpDtype,
     PositionalIndexer,
@@ -889,6 +890,20 @@ def max(self, *, axis: AxisInt | None = None, skipna: bool = True) -> IntervalOr
         indexer = obj.argsort()[-1]
         return obj[indexer]
 
+    def pad_or_backfill(  # pylint: disable=useless-parent-delegation
+        self,
+        *,
+        method: FillnaOptions,
+        limit: int | None = None,
+        limit_area: Literal["inside", "outside"] | None = None,
+        copy: bool = True,
+    ) -> Self:
+        # TODO(3.0): after EA.fillna 'method' deprecation is enforced, we can remove
+        #  this method entirely.
+        return super().pad_or_backfill(
+            method=method, limit=limit, limit_area=limit_area, copy=copy
+        )
+
     def fillna(
         self, value=None, method=None, limit: int | None = None, copy: bool = True
     ) -> Self:
diff --git a/pandas/core/arrays/masked.py b/pandas/core/arrays/masked.py
index 1de3ae3b2428e..bec875f2bbfa1 100644
--- a/pandas/core/arrays/masked.py
+++ b/pandas/core/arrays/masked.py
@@ -24,6 +24,7 @@
     AstypeArg,
     AxisInt,
     DtypeObj,
+    FillnaOptions,
     NpDtype,
     PositionalIndexer,
     Scalar,
@@ -189,6 +190,36 @@ def __getitem__(self, item: PositionalIndexer) -> Self | Any:
 
         return self._simple_new(self._data[item], newmask)
 
+    def pad_or_backfill(
+        self,
+        *,
+        method: FillnaOptions,
+        limit: int | None = None,
+        limit_area: Literal["inside", "outside"] | None = None,
+        copy: bool = True,
+    ) -> Self:
+        mask = self._mask
+
+        if mask.any():
+            func = missing.get_fill_func(method, ndim=self.ndim)
+
+            npvalues = self._data.T
+            new_mask = mask.T
+            if copy:
+                npvalues = npvalues.copy()
+                new_mask = new_mask.copy()
+            func(npvalues, limit=limit, mask=new_mask)
+            if copy:
+                return self._simple_new(npvalues.T, new_mask.T)
+            else:
+                return self
+        else:
+            if copy:
+                new_values = self.copy()
+            else:
+                new_values = self
+        return new_values
+
     @doc(ExtensionArray.fillna)
     def fillna(
         self, value=None, method=None, limit: int | None = None, copy: bool = True
diff --git a/pandas/core/arrays/numpy_.py b/pandas/core/arrays/numpy_.py
index 6d01dfcf6d90b..79a9ffb5f8c0b 100644
--- a/pandas/core/arrays/numpy_.py
+++ b/pandas/core/arrays/numpy_.py
@@ -237,7 +237,7 @@ def pad_or_backfill(
         self,
         *,
         method: FillnaOptions,
-        limit: int | None,
+        limit: int | None = None,
         limit_area: Literal["inside", "outside"] | None = None,
         copy: bool = True,
     ) -> Self:
diff --git a/pandas/core/arrays/period.py b/pandas/core/arrays/period.py
index af6402b9964e5..4df4375c5d701 100644
--- a/pandas/core/arrays/period.py
+++ b/pandas/core/arrays/period.py
@@ -78,6 +78,7 @@
     from pandas._typing import (
         AnyArrayLike,
         Dtype,
+        FillnaOptions,
         NpDtype,
         NumpySorter,
         NumpyValueArrayLike,
@@ -790,6 +791,25 @@ def searchsorted(
         m8arr = self._ndarray.view("M8[ns]")
         return m8arr.searchsorted(npvalue, side=side, sorter=sorter)
 
+    def pad_or_backfill(
+        self,
+        *,
+        method: FillnaOptions,
+        limit: int | None = None,
+        limit_area: Literal["inside", "outside"] | None = None,
+        copy: bool = True,
+    ) -> Self:
+        # view as dt64 so we get treated as timelike in core.missing,
+        #  similar to dtl._period_dispatch
+        dta = self.view("M8[ns]")
+        result = dta.pad_or_backfill(
+            method=method, limit=limit, limit_area=limit_area, copy=copy
+        )
+        if copy:
+            return cast("Self", result.view(self.dtype))
+        else:
+            return self
+
     def fillna(
         self, value=None, method=None, limit: int | None = None, copy: bool = True
     ) -> Self:
diff --git a/pandas/core/arrays/sparse/array.py b/pandas/core/arrays/sparse/array.py
index d832a9f772f45..d32c98535d7cb 100644
--- a/pandas/core/arrays/sparse/array.py
+++ b/pandas/core/arrays/sparse/array.py
@@ -712,6 +712,20 @@ def isna(self):
         mask[self.sp_index.indices] = isna(self.sp_values)
         return type(self)(mask, fill_value=False, dtype=dtype)
 
+    def pad_or_backfill(  # pylint: disable=useless-parent-delegation
+        self,
+        *,
+        method: FillnaOptions,
+        limit: int | None = None,
+        limit_area: Literal["inside", "outside"] | None = None,
+        copy: bool = True,
+    ) -> Self:
+        # TODO(3.0): We can remove this method once deprecation for fillna method
+        #  keyword is enforced.
+        return super().pad_or_backfill(
+            method=method, limit=limit, limit_area=limit_area, copy=copy
+        )
+
     def fillna(
         self,
         value=None,
diff --git a/pandas/core/internals/blocks.py b/pandas/core/internals/blocks.py
index 3e988068dbc12..49f4d333c77b2 100644
--- a/pandas/core/internals/blocks.py
+++ b/pandas/core/internals/blocks.py
@@ -1917,28 +1917,10 @@ def pad_or_backfill(
 
         if values.ndim == 2 and axis == 1:
             # NDArrayBackedExtensionArray.fillna assumes axis=0
-            new_values = values.T.fillna(method=method, limit=limit, copy=copy).T
+            new_values = values.T.pad_or_backfill(method=method, limit=limit).T
         else:
-            try:
-                new_values = values.fillna(method=method, limit=limit, copy=copy)
-            except TypeError:
-                # 3rd party EA that has not implemented copy keyword yet
-                refs = None
-                new_values = values.fillna(method=method, limit=limit)
-                # issue the warning *after* retrying, in case the TypeError
-                #  was caused by an invalid fill_value
-                warnings.warn(
-                    # GH#53278
-                    "ExtensionArray.fillna added a 'copy' keyword in pandas "
-                    "2.1.0. In a future version, ExtensionArray subclasses will "
-                    "need to implement this keyword or an exception will be "
-                    "raised. In the interim, the keyword is ignored by "
-                    f"{type(self.values).__name__}.",
-                    FutureWarning,
-                    stacklevel=find_stack_level(),
-                )
-
-        return [self.make_block_same_class(new_values, refs=refs)]
+            new_values = values.pad_or_backfill(method=method, limit=limit)
+        return [self.make_block_same_class(new_values)]
 
 
 class ExtensionBlock(libinternals.Block, EABackedBlock):
diff --git a/pandas/tests/arrays/test_datetimelike.py b/pandas/tests/arrays/test_datetimelike.py
index 8f87749a4ed6e..711a1b5f2f26c 100644
--- a/pandas/tests/arrays/test_datetimelike.py
+++ b/pandas/tests/arrays/test_datetimelike.py
@@ -258,7 +258,7 @@ def test_fillna_method_doesnt_change_orig(self, method):
 
         fill_value = arr[3] if method == "pad" else arr[5]
 
-        result = arr.fillna(method=method)
+        result = arr.pad_or_backfill(method=method)
         assert result[4] == fill_value
 
         # check that the original was not changed
diff --git a/pandas/tests/arrays/test_datetimes.py b/pandas/tests/arrays/test_datetimes.py
index 1fe1d4efbefd7..8e38a8c741b8d 100644
--- a/pandas/tests/arrays/test_datetimes.py
+++ b/pandas/tests/arrays/test_datetimes.py
@@ -497,7 +497,7 @@ def test_fillna_preserves_tz(self, method):
             dtype=DatetimeTZDtype(tz="US/Central"),
         )
 
-        result = arr.fillna(method=method)
+        result = arr.pad_or_backfill(method=method)
         tm.assert_extension_array_equal(result, expected)
 
         # assert that arr and dti were not modified in-place
@@ -510,12 +510,12 @@ def test_fillna_2d(self):
         dta[0, 1] = pd.NaT
         dta[1, 0] = pd.NaT
 
-        res1 = dta.fillna(method="pad")
+        res1 = dta.pad_or_backfill(method="pad")
         expected1 = dta.copy()
         expected1[1, 0] = dta[0, 0]
         tm.assert_extension_array_equal(res1, expected1)
 
-        res2 = dta.fillna(method="backfill")
+        res2 = dta.pad_or_backfill(method="backfill")
         expected2 = dta.copy()
         expected2 = dta.copy()
         expected2[1, 0] = dta[2, 0]
@@ -529,10 +529,10 @@ def test_fillna_2d(self):
         assert not dta2._ndarray.flags["C_CONTIGUOUS"]
         tm.assert_extension_array_equal(dta, dta2)
 
-        res3 = dta2.fillna(method="pad")
+        res3 = dta2.pad_or_backfill(method="pad")
         tm.assert_extension_array_equal(res3, expected1)
 
-        res4 = dta2.fillna(method="backfill")
+        res4 = dta2.pad_or_backfill(method="backfill")
         tm.assert_extension_array_equal(res4, expected2)
 
         # test the DataFrame method while we're here
diff --git a/pandas/tests/extension/base/dim2.py b/pandas/tests/extension/base/dim2.py
index b9706f87ab7d3..9dcce28f47e52 100644
--- a/pandas/tests/extension/base/dim2.py
+++ b/pandas/tests/extension/base/dim2.py
@@ -155,16 +155,14 @@ def test_concat_2d(self, data):
 
     @pytest.mark.parametrize("method", ["backfill", "pad"])
     def test_fillna_2d_method(self, data_missing, method):
+        # pad_or_backfill is always along axis=0
         arr = data_missing.repeat(2).reshape(2, 2)
         assert arr[0].isna().all()
         assert not arr[1].isna().any()
 
-        try:
-            result = arr.pad_or_backfill(method=method, limit=None)
-        except AttributeError:
-            result = arr.fillna(method=method, limit=None)
+        result = arr.pad_or_backfill(method=method, limit=None)
 
-        expected = data_missing.fillna(method=method).repeat(2).reshape(2, 2)
+        expected = data_missing.pad_or_backfill(method=method).repeat(2).reshape(2, 2)
         self.assert_extension_array_equal(result, expected)
 
         # Reverse so that backfill is not a no-op.
@@ -172,12 +170,11 @@ def test_fillna_2d_method(self, data_missing, method):
         assert not arr2[0].isna().any()
         assert arr2[1].isna().all()
 
-        try:
-            result2 = arr2.pad_or_backfill(method=method, limit=None)
-        except AttributeError:
-            result2 = arr2.fillna(method=method, limit=None)
+        result2 = arr2.pad_or_backfill(method=method, limit=None)
 
-        expected2 = data_missing[::-1].fillna(method=method).repeat(2).reshape(2, 2)
+        expected2 = (
+            data_missing[::-1].pad_or_backfill(method=method).repeat(2).reshape(2, 2)
+        )
         self.assert_extension_array_equal(result2, expected2)
 
     @pytest.mark.parametrize("method", ["mean", "median", "var", "std", "sum", "prod"])
diff --git a/pandas/tests/extension/base/missing.py b/pandas/tests/extension/base/missing.py
index 43f37a020df3f..a839a9d327f95 100644
--- a/pandas/tests/extension/base/missing.py
+++ b/pandas/tests/extension/base/missing.py
@@ -95,7 +95,7 @@ def test_fillna_no_op_returns_copy(self, data):
         assert result is not data
         self.assert_extension_array_equal(result, data)
 
-        result = data.fillna(method="backfill")
+        result = data.pad_or_backfill(method="backfill")
         assert result is not data
         self.assert_extension_array_equal(result, data)
 
diff --git a/pandas/tests/extension/decimal/array.py b/pandas/tests/extension/decimal/array.py
index 5762bc9ce485c..d24b70a884c45 100644
--- a/pandas/tests/extension/decimal/array.py
+++ b/pandas/tests/extension/decimal/array.py
@@ -281,6 +281,8 @@ def convert_values(param):
     def value_counts(self, dropna: bool = True):
         return value_counts(self.to_numpy(), dropna=dropna)
 
+    # We override fillna here to simulate a 3rd party EA that has done so. This
+    #  lets us test the deprecation telling authors to implement pad_or_backfill
     # Simulate a 3rd-party EA that has not yet updated to include a "copy"
     #  keyword in its fillna method.
     # error: Signature of "fillna" incompatible with supertype "ExtensionArray"
diff --git a/pandas/tests/extension/decimal/test_decimal.py b/pandas/tests/extension/decimal/test_decimal.py
index 4f0ff427dd900..6feac7fb9d9dc 100644
--- a/pandas/tests/extension/decimal/test_decimal.py
+++ b/pandas/tests/extension/decimal/test_decimal.py
@@ -105,7 +105,7 @@ def test_fillna_frame(self, data_missing):
             super().test_fillna_frame(data_missing)
 
     def test_fillna_limit_pad(self, data_missing):
-        msg = "ExtensionArray.fillna added a 'copy' keyword"
+        msg = "ExtensionArray.fillna 'method' keyword is deprecated"
         with tm.assert_produces_warning(
             FutureWarning, match=msg, check_stacklevel=False
         ):
@@ -123,6 +123,13 @@ def test_fillna_limit_backfill(self, data_missing):
         ):
             super().test_fillna_limit_backfill(data_missing)
 
+    def test_fillna_no_op_returns_copy(self, data):
+        msg = "ExtensionArray.fillna 'method' keyword is deprecated"
+        with tm.assert_produces_warning(
+            FutureWarning, match=msg, check_stacklevel=False
+        ):
+            super().test_fillna_no_op_returns_copy(data)
+
     def test_fillna_series(self, data_missing):
         msg = "ExtensionArray.fillna added a 'copy' keyword"
         with tm.assert_produces_warning(
@@ -131,7 +138,7 @@ def test_fillna_series(self, data_missing):
             super().test_fillna_series(data_missing)
 
     def test_fillna_series_method(self, data_missing, fillna_method):
-        msg = "ExtensionArray.fillna added a 'copy' keyword"
+        msg = "ExtensionArray.fillna 'method' keyword is deprecated"
         with tm.assert_produces_warning(
             FutureWarning, match=msg, check_stacklevel=False
         ):