diff --git a/Tests/images/mmap_error.bmp b/Tests/images/mmap_error.bmp
new file mode 100644
index 000000000..04df163d7
Binary files /dev/null and b/Tests/images/mmap_error.bmp differ
diff --git a/Tests/test_file_bmp.py b/Tests/test_file_bmp.py
index d58666b44..604d54d88 100644
--- a/Tests/test_file_bmp.py
+++ b/Tests/test_file_bmp.py
@@ -39,6 +39,13 @@ def test_invalid_file():
             BmpImagePlugin.BmpImageFile(fp)
 
 
+def test_fallback_if_mmap_errors():
+    # This image has been truncated,
+    # so that the buffer is not large enough when using mmap
+    with Image.open("Tests/images/mmap_error.bmp") as im:
+        assert_image_equal_tofile(im, "Tests/images/pal8_offset.bmp")
+
+
 def test_save_to_bytes():
     output = io.BytesIO()
     im = hopper()
diff --git a/src/PIL/ImageFile.py b/src/PIL/ImageFile.py
index 9f08493c1..f281b9e14 100644
--- a/src/PIL/ImageFile.py
+++ b/src/PIL/ImageFile.py
@@ -192,6 +192,9 @@ class ImageFile(Image.Image):
 
                     with open(self.filename) as fp:
                         self.map = mmap.mmap(fp.fileno(), 0, access=mmap.ACCESS_READ)
+                    if offset + self.size[1] * args[1] > self.map.size():
+                        # buffer is not large enough
+                        raise OSError
                     self.im = Image.core.map_buffer(
                         self.map, self.size, decoder_name, offset, args
                     )