BUG: Fix dt64[non_nano] + some_offsets incorrectly rounding

pandas/core/arrays/datetimes.py

@@ -64,6 +64,7 @@
 )
 from pandas.core.dtypes.missing import isna
 
+from pandas import Timedelta
 from pandas.core.arrays import datetimelike as dtl
 from pandas.core.arrays._ranges import generate_regular_range
 import pandas.core.common as com
@@ -93,7 +94,6 @@
 
     from pandas import (
         DataFrame,
-        Timedelta,
     )
     from pandas.core.arrays import PeriodArray
 
@@ -817,12 +817,34 @@ def _add_offset(self, offset: BaseOffset) -> Self:
             result = type(self)._from_sequence(res_values, dtype=self.dtype)
 
         else:
-            result = type(self)._simple_new(res_values, dtype=res_values.dtype)
+            units = [
+                "ns",
+                "us",
+                "ms",
+                "s",
+            ]
+            res_unit = self.unit
+            if hasattr(offset, "offset"):
+                offset_td = Timedelta(offset.offset)
+                offset_unit = offset_td.unit
+                if self.unit in units and offset_unit in units:
+                    idx_self = units.index(self.unit)
+                    idx_offset = units.index(offset_unit)
+                    res_unit = units[min(idx_self, idx_offset)]
+            dtype_naive = np.dtype(f"datetime64[{res_unit}]")
+            if res_values.dtype != dtype_naive:
+                res_values = res_values.astype(dtype_naive)
+            result = type(self)._simple_new(res_values, dtype=dtype_naive)
+
             if offset.normalize:
                 result = result.normalize()
                 result._freq = None
 
-            if self.tz is not None:
+            if (
+                self.tz is not None
+                and getattr(result.dtype, "tz", None) is None
+                and res_unit == "ns"
+            ):
                 result = result.tz_localize(self.tz)
 
         return result

pandas/tests/arrays/test_datetimes.py

@@ -844,3 +844,20 @@ def test_factorize_sort_without_freq():
     tda = dta - dta[0]
     with pytest.raises(NotImplementedError, match=msg):
         tda.factorize(sort=True)
+
+
+def test_dt64_non_nano_offset_no_rounding():
+    # GH#56586
+    dti = pd.date_range("2016-01-01", periods=3, unit="s")
+    offset = pd.offsets.CustomBusinessDay(offset=pd.Timedelta("1ms"))
+    result = dti + offset
+
+    assert result.dtype == np.dtype("datetime64[ms]")
+    expected = pd.DatetimeIndex(
+        [
+            pd.Timestamp("2016-01-02 00:00:00.001"),
+            pd.Timestamp("2016-01-03 00:00:00.001"),
+            pd.Timestamp("2016-01-04 00:00:00.001"),
+        ]
+    )
+    tm.assert_index_equal(result, expected)