diff --git a/xarray/core/extension_array.py b/xarray/core/extension_array.py
index d078d61e58a..0c5ef82e043 100644
--- a/xarray/core/extension_array.py
+++ b/xarray/core/extension_array.py
@@ -276,6 +276,14 @@ def __getitem__(self, key) -> PandasExtensionArray[T_ExtensionArray]:
             isinstance(key, tuple) and len(key) == 1
         ):  # pyarrow type arrays can't handle single-length tuples
             (key,) = key
+        # Pandas extension arrays are 1-D and don't support Ellipsis-based
+        # 0-d indexing. NumpyIndexingAdapter appends Ellipsis to force 0-d
+        # slices, but this causes issues with IntervalArray and similar types.
+        # Strip trailing Ellipsis for 1-D extension arrays (GH#11300).
+        if isinstance(key, tuple) and key[-1] is Ellipsis:
+            key = tuple(k for k in key if k is not Ellipsis)
+            if len(key) == 1:
+                (key,) = key
         item = self.array[key]
         if is_allowed_extension_array(item):
             return PandasExtensionArray(item)
diff --git a/xarray/core/variable.py b/xarray/core/variable.py
index 1b5e0d4ff69..c2559893db2 100644
--- a/xarray/core/variable.py
+++ b/xarray/core/variable.py
@@ -836,6 +836,14 @@ def __getitem__(self, key) -> Self:
 
     def _finalize_indexing_result(self, dims, data) -> Self:
         """Used by IndexVariable to return IndexVariable objects when possible."""
+        # PandasExtensionArray is always 1-D and cannot represent scalar (0-d)
+        # data. When indexing produces a scalar result (dims=()), convert to
+        # numpy array (GH#11300).
+        if not dims:
+            from xarray.core.extension_array import PandasExtensionArray
+
+            if isinstance(data, PandasExtensionArray):
+                data = data.array[0]
         return self._replace(dims=dims, data=data)
 
     def _getitem_with_mask(self, key, fill_value=dtypes.NA):
diff --git a/xarray/tests/test_coding_times.py b/xarray/tests/test_coding_times.py
index e03f15a5119..c3eac3d7f00 100644
--- a/xarray/tests/test_coding_times.py
+++ b/xarray/tests/test_coding_times.py
@@ -538,20 +538,21 @@ def test_infer_datetime_units(freq, units) -> None:
 
 
 @pytest.mark.parametrize(
-    ["dates", "expected"],
+    ["date_strings", "expected"],
     [
         (
-            pd.to_datetime(["1900-01-01", "1900-01-02", "NaT"], unit="ns"),
+            ["1900-01-01", "1900-01-02", "NaT"],
             "days since 1900-01-01 00:00:00",
         ),
         (
-            pd.to_datetime(["NaT", "1900-01-01"], unit="ns"),
+            ["NaT", "1900-01-01"],
             "days since 1900-01-01 00:00:00",
         ),
-        (pd.to_datetime(["NaT"], unit="ns"), "days since 1970-01-01 00:00:00"),
+        (["NaT"], "days since 1970-01-01 00:00:00"),
     ],
 )
-def test_infer_datetime_units_with_NaT(dates, expected) -> None:
+def test_infer_datetime_units_with_NaT(date_strings, expected) -> None:
+    dates = pd.to_datetime(date_strings, unit="ns")
     assert expected == infer_datetime_units(dates)
 
 
diff --git a/xarray/tests/test_dataarray.py b/xarray/tests/test_dataarray.py
index 377fd2f8a8b..504b4e51800 100644
--- a/xarray/tests/test_dataarray.py
+++ b/xarray/tests/test_dataarray.py
@@ -6997,6 +6997,28 @@ def test_idxminmax_dask(self, op: str, ndim: int) -> None:
         assert_equal(getattr(ar0_dsk, op)(dim="x"), getattr(ar0_raw, op)(dim="x"))
 
 
+def test_idxminmax_interval_coords() -> None:
+    # GH#11300 - idxmax/idxmin should work with IntervalIndex coordinates
+    import pandas as pd
+
+    idx = pd.IntervalIndex.from_breaks([0, 1, 2, 3])
+    da = xr.DataArray([False, True, True], dims=["z"], coords={"z": idx})
+
+    result = da.idxmax()
+    assert result.values.item() == pd.Interval(1, 2, closed="right")
+
+    result = da.idxmin()
+    assert result.values.item() == pd.Interval(0, 1, closed="right")
+
+    # Test with skipna=False
+    da2 = xr.DataArray([np.nan, 1.0, 2.0], dims=["z"], coords={"z": idx})
+    result2 = da2.idxmax()
+    assert result2.values.item() == pd.Interval(2, 3, closed="right")
+
+    result3 = da2.idxmin()
+    assert result3.values.item() == pd.Interval(1, 2, closed="right")
+
+
 @pytest.mark.parametrize("da", ("repeating_ints",), indirect=True)
 def test_isin(da) -> None:
     expected = DataArray(