diff --git a/spacy/cli/_util.py b/spacy/cli/_util.py
index ba3892b1d96..3f536a2128b 100644
--- a/spacy/cli/_util.py
+++ b/spacy/cli/_util.py
@@ -414,7 +414,10 @@ def git_checkout(
             if not is_subpath_of(tmp_dir, source_path):
                 err = f"'{subpath}' is a path outside of the cloned repository."
                 msg.fail(err, repo, exits=1)
-            shutil.copytree(str(source_path), str(dest))
+            if os.path.isdir(source_path):
+                shutil.copytree(str(source_path), str(dest))
+            else:
+                shutil.copyfile(source_path, dest)
         except FileNotFoundError:
             err = f"Can't clone {subpath}. Make sure the directory exists in the repo (branch '{branch}')"
             msg.fail(err, repo, exits=1)
diff --git a/spacy/tests/test_cli.py b/spacy/tests/test_cli.py
index 249c446727f..db34e6f9f0a 100644
--- a/spacy/tests/test_cli.py
+++ b/spacy/tests/test_cli.py
@@ -20,7 +20,7 @@
 from spacy.cli._util import parse_config_overrides, string_to_list
 from spacy.cli._util import substitute_project_variables
 from spacy.cli._util import validate_project_commands
-from spacy.cli._util import upload_file, download_file
+from spacy.cli._util import upload_file, download_file, git_checkout
 from spacy.cli.debug_data import _compile_gold, _get_labels_from_model
 from spacy.cli.debug_data import _get_labels_from_spancat
 from spacy.cli.debug_data import _get_distribution, _get_kl_divergence
@@ -145,6 +145,31 @@ def test_issue11235():
     assert cfg["commands"][0]["script"][0] == f"hello {lang_var}"
 
 
+def test_project_git_dir_asset():
+    with make_tempdir() as d:
+        # Use a very small repo.
+        git_checkout(
+            "https://github.com/explosion/os-signpost.git",
+            "os_signpost",
+            d / "signpost",
+            branch="v0.0.3",
+        )
+        assert os.path.isdir(d / "signpost")
+
+
+@pytest.mark.issue(12168)
+def test_project_git_file_asset():
+    with make_tempdir() as d:
+        # Use a very small repo.
+        git_checkout(
+            "https://github.com/explosion/os-signpost.git",
+            "README.md",
+            d / "readme.md",
+            branch="v0.0.3",
+        )
+        assert os.path.isfile(d / "readme.md")
+
+
 def test_cli_info():
     nlp = Dutch()
     nlp.add_pipe("textcat")