diff --git a/pandas/core/arrays/_mixins.py b/pandas/core/arrays/_mixins.py
index d1f8957859337..832d09b062265 100644
--- a/pandas/core/arrays/_mixins.py
+++ b/pandas/core/arrays/_mixins.py
@@ -4,6 +4,7 @@
 
 from pandas.compat.numpy import function as nv
 from pandas.errors import AbstractMethodError
+from pandas.util._decorators import cache_readonly
 
 from pandas.core.algorithms import take, unique
 from pandas.core.arrays.base import ExtensionArray
@@ -64,6 +65,8 @@ def _validate_fill_value(self, fill_value):
 
     # ------------------------------------------------------------------------
 
+    # TODO: make this a cache_readonly; for that to work we need to remove
+    #  the _index_data kludge in libreduction
     @property
     def shape(self) -> Tuple[int, ...]:
         return self._ndarray.shape
@@ -71,15 +74,15 @@ def shape(self) -> Tuple[int, ...]:
     def __len__(self) -> int:
         return self.shape[0]
 
-    @property
+    @cache_readonly
     def ndim(self) -> int:
         return len(self.shape)
 
-    @property
+    @cache_readonly
     def size(self) -> int:
         return np.prod(self.shape)
 
-    @property
+    @cache_readonly
     def nbytes(self) -> int:
         return self._ndarray.nbytes
 
diff --git a/pandas/core/arrays/datetimelike.py b/pandas/core/arrays/datetimelike.py
index e07e2da164cac..257a51b423308 100644
--- a/pandas/core/arrays/datetimelike.py
+++ b/pandas/core/arrays/datetimelike.py
@@ -454,6 +454,8 @@ class DatetimeLikeArrayMixin(
     # ------------------------------------------------------------------
     # NDArrayBackedExtensionArray compat
 
+    # TODO: make this a cache_readonly; need to get around _index_data
+    #  kludge in libreduction
     @property
     def _ndarray(self) -> np.ndarray:
         # NB: A bunch of Interval tests fail if we use ._data
@@ -526,6 +528,13 @@ def __getitem__(self, key):
         only handle list-likes, slices, and integer scalars
         """
 
+        if lib.is_integer(key):
+            # fast-path
+            result = self._data[key]
+            if self.ndim == 1:
+                return self._box_func(result)
+            return self._simple_new(result, dtype=self.dtype)
+
         if com.is_bool_indexer(key):
             # first convert to boolean, because check_array_indexer doesn't
             # allow object dtype