Android 4.4 Google Play Services Apk ~upd~ Today

While the official Play Store might not serve you the latest version, you can still find the final compatible APKs (versions up to roughly ) through community repositories. APKMirror (Android 4.4+ Library)

If you are looking to side-load or restore functionality, you can find archived versions on reputable community-driven platforms: android 4.4 google play services apk