diff --git a/doc/whats-new.rst b/doc/whats-new.rst
index 4995b2953f9..5e0c6825a15 100644
--- a/doc/whats-new.rst
+++ b/doc/whats-new.rst
@@ -71,6 +71,8 @@ Bug fixes
   had no effect. (Mentioned in :issue:`9921`)
 - Enable ``keep_attrs`` in ``DatasetView.map`` relevant for :py:func:`map_over_datasets`  (:pull:`10219`)
   By `Mathias Hauser <https://github.com/mathause>`_.
+- Variables with no temporal dimension are left untouched by :py:meth:`~xarray.Dataset.convert_calendar`. (:issue:`10266`,  :pull:`10268`)
+  By `Pascal Bourgault <https://github.com/aulemahal>`_.
 
 Documentation
 ~~~~~~~~~~~~~
diff --git a/xarray/coding/calendar_ops.py b/xarray/coding/calendar_ops.py
index ec5a7efba4b..5fdd106e179 100644
--- a/xarray/coding/calendar_ops.py
+++ b/xarray/coding/calendar_ops.py
@@ -213,7 +213,7 @@ def convert_calendar(
         out[dim] = new_times
 
         # Remove NaN that where put on invalid dates in target calendar
-        out = out.where(out[dim].notnull(), drop=True)
+        out = out.sel(time=out[dim].notnull())
 
         if use_cftime:
             # Reassign times to ensure time index of output is a CFTimeIndex
diff --git a/xarray/tests/test_calendar_ops.py b/xarray/tests/test_calendar_ops.py
index 13e9f7a1030..8dc1c2a503b 100644
--- a/xarray/tests/test_calendar_ops.py
+++ b/xarray/tests/test_calendar_ops.py
@@ -4,7 +4,7 @@
 import pandas as pd
 import pytest
 
-from xarray import CFTimeIndex, DataArray, infer_freq
+from xarray import CFTimeIndex, DataArray, Dataset, infer_freq
 from xarray.coding.calendar_ops import convert_calendar, interp_calendar
 from xarray.coding.cftime_offsets import date_range
 from xarray.testing import assert_identical
@@ -63,6 +63,24 @@ def test_convert_calendar(source, target, use_cftime, freq):
     np.testing.assert_array_equal(conv.time, expected_times)
 
 
+def test_convert_calendar_dataset():
+    # Check that variables without a time dimension are not modified
+    src = DataArray(
+        date_range("2004-01-01", "2004-12-31", freq="D", calendar="standard"),
+        dims=("time",),
+        name="time",
+    )
+    da_src = DataArray(
+        np.linspace(0, 1, src.size), dims=("time",), coords={"time": src}
+    ).expand_dims(lat=[0, 1])
+    ds_src = Dataset({"hastime": da_src, "notime": (("lat",), [0, 1])})
+
+    conv = convert_calendar(ds_src, "360_day", align_on="date")
+
+    assert conv.time.dt.calendar == "360_day"
+    assert_identical(ds_src.notime, conv.notime)
+
+
 @pytest.mark.parametrize(
     "source,target,freq",
     [