diff --git a/doc/source/whatsnew/v2.2.0.rst b/doc/source/whatsnew/v2.2.0.rst
index 5b955aa45219a..2fcab46c9e229 100644
--- a/doc/source/whatsnew/v2.2.0.rst
+++ b/doc/source/whatsnew/v2.2.0.rst
@@ -727,6 +727,7 @@ Timezones
 Numeric
 ^^^^^^^
 - Bug in :func:`read_csv` with ``engine="pyarrow"`` causing rounding errors for large integers (:issue:`52505`)
+- Bug in :meth:`Series.__floordiv__` for :class:`ArrowDtype` with integral dtypes raising for large values (:issue:`56645`)
 - Bug in :meth:`Series.pow` not filling missing values correctly (:issue:`55512`)
 
 Conversion
diff --git a/pandas/core/arrays/arrow/array.py b/pandas/core/arrays/arrow/array.py
index de1ed9ecfdaf1..59f0a3af2b1ab 100644
--- a/pandas/core/arrays/arrow/array.py
+++ b/pandas/core/arrays/arrow/array.py
@@ -114,7 +114,12 @@ def cast_for_truediv(
         if pa.types.is_integer(arrow_array.type) and pa.types.is_integer(
             pa_object.type
         ):
-            return arrow_array.cast(pa.float64())
+            # https://github.com/apache/arrow/issues/35563
+            # Arrow does not allow safe casting large integral values to float64.
+            # Intentionally not using arrow_array.cast because it could be a scalar
+            # value in reflected case, and safe=False only added to
+            # scalar cast in pyarrow 13.
+            return pc.cast(arrow_array, pa.float64(), safe=False)
         return arrow_array
 
     def floordiv_compat(
diff --git a/pandas/tests/extension/test_arrow.py b/pandas/tests/extension/test_arrow.py
index 5624acfb64764..643a42d32ebe2 100644
--- a/pandas/tests/extension/test_arrow.py
+++ b/pandas/tests/extension/test_arrow.py
@@ -3133,6 +3133,14 @@ def test_arrow_floordiv():
     tm.assert_series_equal(result, expected)
 
 
+def test_arrow_floordiv_large_values():
+    # GH 55561
+    a = pd.Series([1425801600000000000], dtype="int64[pyarrow]")
+    expected = pd.Series([1425801600000], dtype="int64[pyarrow]")
+    result = a // 1_000_000
+    tm.assert_series_equal(result, expected)
+
+
 def test_string_to_datetime_parsing_cast():
     # GH 56266
     string_dates = ["2020-01-01 04:30:00", "2020-01-02 00:00:00", "2020-01-03 00:00:00"]