diff --git a/doc/source/whatsnew/v1.4.4.rst b/doc/source/whatsnew/v1.4.4.rst
index 96e4ad4321c60..5fcd27dd39884 100644
--- a/doc/source/whatsnew/v1.4.4.rst
+++ b/doc/source/whatsnew/v1.4.4.rst
@@ -32,6 +32,7 @@ Bug fixes
 - The :class:`errors.FutureWarning` raised when passing arguments (other than ``filepath_or_buffer``) as positional in :func:`read_csv` is now raised at the correct stacklevel (:issue:`47385`)
 - Bug in :meth:`DataFrame.to_sql` when ``method`` was a ``callable`` that did not return an ``int`` and would raise a ``TypeError`` (:issue:`46891`)
 - Bug in :meth:`loc.__getitem__` with a list of keys causing an internal inconsistency that could lead to a disconnect between ``frame.at[x, y]`` vs ``frame[y].loc[x]`` (:issue:`22372`)
+- Bug in the :meth:`Series.dt.strftime` accessor return a float instead of object dtype Series for all-NaT input, which also causes a spurious deprecation warning (:issue:`45858`)
 
 .. ---------------------------------------------------------------------------
 
diff --git a/pandas/core/indexes/datetimes.py b/pandas/core/indexes/datetimes.py
index 004d860b20a6f..4f93db4499a30 100644
--- a/pandas/core/indexes/datetimes.py
+++ b/pandas/core/indexes/datetimes.py
@@ -110,7 +110,7 @@ def _new_DatetimeIndex(cls, d):
     + [
         method
         for method in DatetimeArray._datetimelike_methods
-        if method not in ("tz_localize", "tz_convert")
+        if method not in ("tz_localize", "tz_convert", "strftime")
     ],
     DatetimeArray,
     wrap=True,
@@ -261,7 +261,7 @@ class DatetimeIndex(DatetimeTimedeltaMixin):
     @doc(DatetimeArray.strftime)
     def strftime(self, date_format) -> Index:
         arr = self._data.strftime(date_format)
-        return Index(arr, name=self.name)
+        return Index(arr, name=self.name, dtype=object)
 
     @doc(DatetimeArray.tz_convert)
     def tz_convert(self, tz) -> DatetimeIndex:
diff --git a/pandas/tests/series/accessors/test_dt_accessor.py b/pandas/tests/series/accessors/test_dt_accessor.py
index 41b5e55e75213..fe4b25026ac74 100644
--- a/pandas/tests/series/accessors/test_dt_accessor.py
+++ b/pandas/tests/series/accessors/test_dt_accessor.py
@@ -618,6 +618,17 @@ def test_strftime_nat(self, data):
         expected = Series(["2019-01-01", np.nan])
         tm.assert_series_equal(result, expected)
 
+    @pytest.mark.parametrize(
+        "data", [DatetimeIndex([pd.NaT]), PeriodIndex([pd.NaT], dtype="period[D]")]
+    )
+    def test_strftime_all_nat(self, data):
+        # https://github.com/pandas-dev/pandas/issues/45858
+        ser = Series(data)
+        with tm.assert_produces_warning(None):
+            result = ser.dt.strftime("%Y-%m-%d")
+        expected = Series([np.nan], dtype=object)
+        tm.assert_series_equal(result, expected)
+
     def test_valid_dt_with_missing_values(self):
 
         from datetime import (