diff --git a/pandas/core/algorithms.py b/pandas/core/algorithms.py
index 1c2e55c3e37df..db45f140c268e 100644
--- a/pandas/core/algorithms.py
+++ b/pandas/core/algorithms.py
@@ -1673,7 +1673,10 @@ def union_with_duplicates(
 
 
 def map_array(
-    arr: ArrayLike, mapper, na_action: Literal["ignore"] | None = None
+    arr: ArrayLike,
+    mapper,
+    na_action: Literal["ignore"] | None = None,
+    convert: bool = True,
 ) -> np.ndarray | ExtensionArray | Index:
     """
     Map values using an input mapping or function.
@@ -1685,6 +1688,9 @@ def map_array(
     na_action : {None, 'ignore'}, default None
         If 'ignore', propagate NA values, without passing them to the
         mapping correspondence.
+    convert : bool, default True
+        Try to find better dtype for elementwise function results. If
+        False, leave as dtype=object.
 
     Returns
     -------
@@ -1739,6 +1745,8 @@ def map_array(
     # we must convert to python types
     values = arr.astype(object, copy=False)
     if na_action is None:
-        return lib.map_infer(values, mapper)
+        return lib.map_infer(values, mapper, convert=convert)
     else:
-        return lib.map_infer_mask(values, mapper, isna(values).view(np.uint8))
+        return lib.map_infer_mask(
+            values, mapper, mask=isna(values).view(np.uint8), convert=convert
+        )
diff --git a/pandas/core/apply.py b/pandas/core/apply.py
index f23b05a5c41b8..65fe9144618ee 100644
--- a/pandas/core/apply.py
+++ b/pandas/core/apply.py
@@ -24,7 +24,6 @@
 
 from pandas._config import option_context
 
-from pandas._libs import lib
 from pandas._typing import (
     AggFuncType,
     AggFuncTypeBase,
@@ -1063,20 +1062,12 @@ def apply_standard(self) -> DataFrame | Series:
         f = cast(Callable, self.f)
         obj = self.obj
 
-        with np.errstate(all="ignore"):
-            if isinstance(f, np.ufunc):
+        if isinstance(f, np.ufunc):
+            with np.errstate(all="ignore"):
                 return f(obj)
 
-            # row-wise access
-            if is_extension_array_dtype(obj.dtype):
-                mapped = obj._values.map(f)
-            else:
-                values = obj.astype(object)._values
-                mapped = lib.map_infer(
-                    values,
-                    f,
-                    convert=self.convert_dtype,
-                )
+        # row-wise access
+        mapped = obj._map_values(mapper=f, convert=self.convert_dtype)
 
         if len(mapped) and isinstance(mapped[0], ABCSeries):
             # GH#43986 Need to do list(mapped) in order to get treated as nested
diff --git a/pandas/core/base.py b/pandas/core/base.py
index 2177b8f3d58a3..dc37550ced387 100644
--- a/pandas/core/base.py
+++ b/pandas/core/base.py
@@ -865,7 +865,7 @@ def _reduce(
         return func(skipna=skipna, **kwds)
 
     @final
-    def _map_values(self, mapper, na_action=None):
+    def _map_values(self, mapper, na_action=None, convert: bool = True):
         """
         An internal function that maps values using the input
         correspondence (which can be a dict, Series, or function).
@@ -877,6 +877,10 @@ def _map_values(self, mapper, na_action=None):
         na_action : {None, 'ignore'}
             If 'ignore', propagate NA values, without passing them to the
             mapping function
+        convert : bool, default True
+            Try to find better dtype for elementwise function results. If
+            False, leave as dtype=object. Note that the dtype is always
+            preserved for some extension array dtypes, such as Categorical.
 
         Returns
         -------
@@ -894,7 +898,7 @@ def _map_values(self, mapper, na_action=None):
         # "Union[IndexOpsMixin, ExtensionArray, ndarray[Any, Any]]";
         # expected "Union[ExtensionArray, ndarray[Any, Any]]"
         return algorithms.map_array(
-            arr, mapper, na_action=na_action  # type: ignore[arg-type]
+            arr, mapper, na_action=na_action, convert=convert  # type: ignore[arg-type]
         )
 
     @final