diff --git a/pandas/conftest.py b/pandas/conftest.py
index 3fdde3261bd68..35affa62ccf68 100644
--- a/pandas/conftest.py
+++ b/pandas/conftest.py
@@ -1144,6 +1144,8 @@ def nullable_string_dtype(request):
     * 'string'
     * 'arrow_string'
     """
+    from pandas.core.arrays.string_arrow import ArrowStringDtype  # noqa: F401
+
     return request.param
 
 
diff --git a/pandas/tests/frame/methods/test_astype.py b/pandas/tests/frame/methods/test_astype.py
index 161fe7990a327..544960113fafc 100644
--- a/pandas/tests/frame/methods/test_astype.py
+++ b/pandas/tests/frame/methods/test_astype.py
@@ -3,6 +3,8 @@
 import numpy as np
 import pytest
 
+import pandas.util._test_decorators as td
+
 import pandas as pd
 from pandas import (
     Categorical,
@@ -564,17 +566,25 @@ def test_astype_empty_dtype_dict(self):
         assert result is not df
 
     @pytest.mark.parametrize(
-        "df",
+        "data, dtype",
         [
-            DataFrame(Series(["x", "y", "z"], dtype="string")),
-            DataFrame(Series(["x", "y", "z"], dtype="category")),
-            DataFrame(Series(3 * [Timestamp("2020-01-01", tz="UTC")])),
-            DataFrame(Series(3 * [Interval(0, 1)])),
+            (["x", "y", "z"], "string"),
+            pytest.param(
+                ["x", "y", "z"],
+                "arrow_string",
+                marks=td.skip_if_no("pyarrow", min_version="1.0.0"),
+            ),
+            (["x", "y", "z"], "category"),
+            (3 * [Timestamp("2020-01-01", tz="UTC")], None),
+            (3 * [Interval(0, 1)], None),
         ],
     )
     @pytest.mark.parametrize("errors", ["raise", "ignore"])
-    def test_astype_ignores_errors_for_extension_dtypes(self, df, errors):
+    def test_astype_ignores_errors_for_extension_dtypes(self, data, dtype, errors):
         # https://github.com/pandas-dev/pandas/issues/35471
+        from pandas.core.arrays.string_arrow import ArrowStringDtype  # noqa: F401
+
+        df = DataFrame(Series(data, dtype=dtype))
         if errors == "ignore":
             expected = df
             result = df.astype(float, errors=errors)
diff --git a/pandas/tests/frame/methods/test_select_dtypes.py b/pandas/tests/frame/methods/test_select_dtypes.py
index 7d3333e493136..3ff1ceba7996b 100644
--- a/pandas/tests/frame/methods/test_select_dtypes.py
+++ b/pandas/tests/frame/methods/test_select_dtypes.py
@@ -391,7 +391,6 @@ def test_select_dtypes_typecodes(self):
         (
             (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),
         ),
@@ -402,3 +401,9 @@ def test_select_dtypes_numeric(self, arr, expected):
         df = DataFrame(arr)
         is_selected = df.select_dtypes(np.number).shape == df.shape
         assert is_selected == expected
+
+    def test_select_dtypes_numeric_nullable_string(self, nullable_string_dtype):
+        arr = pd.array(["a", "b"], dtype=nullable_string_dtype)
+        df = DataFrame(arr)
+        is_selected = df.select_dtypes(np.number).shape == df.shape
+        assert not is_selected
diff --git a/pandas/tests/indexing/test_check_indexer.py b/pandas/tests/indexing/test_check_indexer.py
index 0e52c075d5af5..975a31b873792 100644
--- a/pandas/tests/indexing/test_check_indexer.py
+++ b/pandas/tests/indexing/test_check_indexer.py
@@ -78,7 +78,6 @@ def test_int_raise_missing_values(indexer):
         np.array([1.0, 2.0], dtype="float64"),
         np.array([True, False], dtype=object),
         pd.Index([True, False], dtype=object),
-        pd.array(["a", "b"], dtype="string"),
     ],
 )
 def test_raise_invalid_array_dtypes(indexer):
@@ -89,6 +88,15 @@ def test_raise_invalid_array_dtypes(indexer):
         check_array_indexer(arr, indexer)
 
 
+def test_raise_nullable_string_dtype(nullable_string_dtype):
+    indexer = pd.array(["a", "b"], dtype=nullable_string_dtype)
+    arr = np.array([1, 2, 3])
+
+    msg = "arrays used as indices must be of integer or boolean type"
+    with pytest.raises(IndexError, match=msg):
+        check_array_indexer(arr, indexer)
+
+
 @pytest.mark.parametrize("indexer", [None, Ellipsis, slice(0, 3), (None,)])
 def test_pass_through_non_array_likes(indexer):
     arr = np.array([1, 2, 3])