diff --git a/pandas/core/arrays/_mixins.py b/pandas/core/arrays/_mixins.py
index 4615cb4ec7abd..d54d1855ac2f8 100644
--- a/pandas/core/arrays/_mixins.py
+++ b/pandas/core/arrays/_mixins.py
@@ -278,8 +278,12 @@ def fillna(
 
         if mask.any():
             if method is not None:
-                func = missing.get_fill_func(method)
-                new_values, _ = func(self._ndarray.copy(), limit=limit, mask=mask)
+                # TODO: check value is None
+                # (for now) when self.ndim == 2, we assume axis=0
+                func = missing.get_fill_func(method, ndim=self.ndim)
+                new_values, _ = func(self._ndarray.T.copy(), limit=limit, mask=mask.T)
+                new_values = new_values.T
+
                 # TODO: PandasArray didn't used to copy, need tests for this
                 new_values = self._from_backing_data(new_values)
             else:
diff --git a/pandas/core/arrays/period.py b/pandas/core/arrays/period.py
index 96a159c0804c9..7e9e13400e11f 100644
--- a/pandas/core/arrays/period.py
+++ b/pandas/core/arrays/period.py
@@ -639,6 +639,14 @@ def searchsorted(self, value, side="left", sorter=None) -> np.ndarray:
         m8arr = self._ndarray.view("M8[ns]")
         return m8arr.searchsorted(value, side=side, sorter=sorter)
 
+    def fillna(self, value=None, method=None, limit=None) -> PeriodArray:
+        if method is not None:
+            # view as dt64 so we get treated as timelike in core.missing
+            dta = self.view("M8[ns]")
+            result = dta.fillna(value=value, method=method, limit=limit)
+            return result.view(self.dtype)
+        return super().fillna(value=value, method=method, limit=limit)
+
     # ------------------------------------------------------------------
     # Arithmetic Methods
 
diff --git a/pandas/core/missing.py b/pandas/core/missing.py
index 1b5a7237b5287..dc42a175409c2 100644
--- a/pandas/core/missing.py
+++ b/pandas/core/missing.py
@@ -646,8 +646,6 @@ def interpolate_2d(
             values,
         )
 
-    orig_values = values
-
     transf = (lambda x: x) if axis == 0 else (lambda x: x.T)
 
     # reshape a 1 dim if needed
@@ -669,10 +667,6 @@ def interpolate_2d(
     if ndim == 1:
         result = result[0]
 
-    if orig_values.dtype.kind in ["m", "M"]:
-        # convert float back to datetime64/timedelta64
-        result = result.view(orig_values.dtype)
-
     return result
 
 
@@ -755,9 +749,11 @@ def _backfill_2d(values, limit=None, mask=None):
 _fill_methods = {"pad": _pad_1d, "backfill": _backfill_1d}
 
 
-def get_fill_func(method):
+def get_fill_func(method, ndim: int = 1):
     method = clean_fill_method(method)
-    return _fill_methods[method]
+    if ndim == 1:
+        return _fill_methods[method]
+    return {"pad": _pad_2d, "backfill": _backfill_2d}[method]
 
 
 def clean_reindex_fill_method(method):
diff --git a/pandas/tests/arrays/test_datetimes.py b/pandas/tests/arrays/test_datetimes.py
index d159d76030250..8e6c330475e68 100644
--- a/pandas/tests/arrays/test_datetimes.py
+++ b/pandas/tests/arrays/test_datetimes.py
@@ -195,6 +195,37 @@ def test_fillna_preserves_tz(self, method):
         assert arr[2] is pd.NaT
         assert dti[2] == pd.Timestamp("2000-01-03", tz="US/Central")
 
+    def test_fillna_2d(self):
+        dti = pd.date_range("2016-01-01", periods=6, tz="US/Pacific")
+        dta = dti._data.reshape(3, 2).copy()
+        dta[0, 1] = pd.NaT
+        dta[1, 0] = pd.NaT
+
+        res1 = dta.fillna(method="pad")
+        expected1 = dta.copy()
+        expected1[1, 0] = dta[0, 0]
+        tm.assert_extension_array_equal(res1, expected1)
+
+        res2 = dta.fillna(method="backfill")
+        expected2 = dta.copy()
+        expected2 = dta.copy()
+        expected2[1, 0] = dta[2, 0]
+        expected2[0, 1] = dta[1, 1]
+        tm.assert_extension_array_equal(res2, expected2)
+
+        # with different ordering for underlying ndarray; behavior should
+        #  be unchanged
+        dta2 = dta._from_backing_data(dta._ndarray.copy(order="F"))
+        assert dta2._ndarray.flags["F_CONTIGUOUS"]
+        assert not dta2._ndarray.flags["C_CONTIGUOUS"]
+        tm.assert_extension_array_equal(dta, dta2)
+
+        res3 = dta2.fillna(method="pad")
+        tm.assert_extension_array_equal(res3, expected1)
+
+        res4 = dta2.fillna(method="backfill")
+        tm.assert_extension_array_equal(res4, expected2)
+
     def test_array_interface_tz(self):
         tz = "US/Central"
         data = DatetimeArray(pd.date_range("2017", periods=2, tz=tz))
diff --git a/pandas/tests/extension/base/dim2.py b/pandas/tests/extension/base/dim2.py
index fbe2537e8a7bf..073880d79d872 100644
--- a/pandas/tests/extension/base/dim2.py
+++ b/pandas/tests/extension/base/dim2.py
@@ -131,6 +131,17 @@ def test_concat_2d(self, data):
         with pytest.raises(ValueError):
             left._concat_same_type([left, right], axis=2)
 
+    @pytest.mark.parametrize("method", ["backfill", "pad"])
+    def test_fillna_2d_method(self, data_missing, method):
+        arr = data_missing.repeat(2).reshape(2, 2)
+        assert arr[0].isna().all()
+        assert not arr[1].isna().any()
+
+        result = arr.fillna(method=method)
+
+        expected = data_missing.fillna(method=method).repeat(2).reshape(2, 2)
+        self.assert_extension_array_equal(result, expected)
+
     @pytest.mark.parametrize("method", ["mean", "median", "var", "std", "sum", "prod"])
     def test_reductions_2d_axis_none(self, data, method, request):
         if not hasattr(data, method):