BUG: Fixing DataFrame.Update crashes when NaT present

pandas/core/frame.py

@@ -8190,11 +8190,16 @@ def update(
         if not isinstance(other, DataFrame):
             other = DataFrame(other)
 
-        other = other.reindex_like(self)
+        # reindex rows, non-matching columns get skipped
+        other = other.reindex(self.index)
 
         for col in self.columns:
+            if col not in other.columns:
+                continue
+
             this = self[col]._values
             that = other[col]._values
+
             if filter_func is not None:
                 with np.errstate(all="ignore"):
                     mask = ~filter_func(this) | isna(that)

pandas/tests/frame/methods/test_update.py

@@ -166,3 +166,37 @@ def test_update_modify_view(self, using_copy_on_write):
             tm.assert_frame_equal(result_view, df2_orig)
         else:
             tm.assert_frame_equal(result_view, expected)
+
+    def test_update_dt_column_with_NaT_create_column(self):
+        df = DataFrame(
+            {
+                "A": [1, None],
+                "B": [
+                    pd.NaT,
+                    pd.to_datetime("2016-01-01"),
+                ],
+            }
+        )
+        df2 = DataFrame({"A": [2, 3]})
+
+        df.update(df2, overwrite=False)
+
+        expected = DataFrame(
+            {"A": [1.0, 3.0], "B": [pd.NaT, pd.to_datetime("2016-01-01")]}
+        )
+
+        tm.assert_frame_equal(df, expected)
+
+    def test_update_dt_column_with_NaT_create_row(self):
+
+        df = DataFrame({"A": [1, None], "B": [pd.to_datetime("2017-1-1"), pd.NaT]})
+
+        df2 = DataFrame({"A": [2], "B": [pd.to_datetime("2016-01-01")]})
+
+        df.update(df2, overwrite=False)
+
+        expected = DataFrame(
+            {"A": [1, None], "B": [pd.to_datetime("2017-1-1"), pd.NaT]}
+        )
+
+        tm.assert_frame_equal(df, expected)