diff --git a/doc/source/whatsnew/v1.3.0.rst b/doc/source/whatsnew/v1.3.0.rst
index 05d3b1c797375..020d3091929db 100644
--- a/doc/source/whatsnew/v1.3.0.rst
+++ b/doc/source/whatsnew/v1.3.0.rst
@@ -174,7 +174,7 @@ Timezones
 Numeric
 ^^^^^^^
 - Bug in :meth:`DataFrame.quantile`, :meth:`DataFrame.sort_values` causing incorrect subsequent indexing behavior (:issue:`38351`)
--
+- Bug in :meth:`DataFrame.select_dtypes` with ``include=np.number`` now retains numeric ``ExtensionDtype`` columns (:issue:`35340`)
 -
 
 Conversion
diff --git a/pandas/core/frame.py b/pandas/core/frame.py
index d698edbb0b8ad..d123e0735ded1 100644
--- a/pandas/core/frame.py
+++ b/pandas/core/frame.py
@@ -3719,12 +3719,14 @@ def extract_unique_dtypes_from_dtypes_set(
             extracted_dtypes = [
                 unique_dtype
                 for unique_dtype in unique_dtypes
-                # error: Argument 1 to "tuple" has incompatible type
-                # "FrozenSet[Union[ExtensionDtype, str, Any, Type[str],
-                # Type[float], Type[int], Type[complex], Type[bool]]]";
-                # expected "Iterable[Union[type, Tuple[Any, ...]]]"
-                if issubclass(
-                    unique_dtype.type, tuple(dtypes_set)  # type: ignore[arg-type]
+                if (
+                    issubclass(
+                        unique_dtype.type, tuple(dtypes_set)  # type: ignore[arg-type]
+                    )
+                    or (
+                        np.number in dtypes_set
+                        and getattr(unique_dtype, "_is_numeric", False)
+                    )
                 )
             ]
             return extracted_dtypes
diff --git a/pandas/tests/frame/methods/test_select_dtypes.py b/pandas/tests/frame/methods/test_select_dtypes.py
index 2a8826cedd50a..f2dbe4a799a17 100644
--- a/pandas/tests/frame/methods/test_select_dtypes.py
+++ b/pandas/tests/frame/methods/test_select_dtypes.py
@@ -1,9 +1,46 @@
 import numpy as np
 import pytest
 
+from pandas.core.dtypes.dtypes import ExtensionDtype
+
 import pandas as pd
 from pandas import DataFrame, Timestamp
 import pandas._testing as tm
+from pandas.core.arrays import ExtensionArray
+
+
+class DummyDtype(ExtensionDtype):
+    type = int
+
+    def __init__(self, numeric):
+        self._numeric = numeric
+
+    @property
+    def name(self):
+        return "Dummy"
+
+    @property
+    def _is_numeric(self):
+        return self._numeric
+
+
+class DummyArray(ExtensionArray):
+    def __init__(self, data, dtype):
+        self.data = data
+        self._dtype = dtype
+
+    def __array__(self, dtype):
+        return self.data
+
+    @property
+    def dtype(self):
+        return self._dtype
+
+    def __len__(self) -> int:
+        return len(self.data)
+
+    def __getitem__(self, item):
+        pass
 
 
 class TestSelectDtypes:
@@ -322,3 +359,20 @@ def test_select_dtypes_typecodes(self):
         expected = df
         FLOAT_TYPES = list(np.typecodes["AllFloat"])
         tm.assert_frame_equal(df.select_dtypes(FLOAT_TYPES), expected)
+
+    @pytest.mark.parametrize(
+        "arr,expected",
+        (
+            (np.array([1, 2], dtype=np.int32), True),
+            (pd.array([1, 2], dtype="Int32"), True),
+            (pd.array(["a", "b"], dtype="string"), False),
+            (DummyArray([1, 2], dtype=DummyDtype(numeric=True)), True),
+            (DummyArray([1, 2], dtype=DummyDtype(numeric=False)), False),
+        ),
+    )
+    def test_select_dtypes_numeric(self, arr, expected):
+        # GH 35340
+
+        df = DataFrame(arr)
+        is_selected = df.select_dtypes(np.number).shape == df.shape
+        assert is_selected == expected