If you prefer not to sideload, you can force a check for the latest compatible version directly in the app: Open the Play Store. Profile Icon (top right) > Update Play Store
The latest version of the Google Play Store APK for Android 5.1.1 is (or higher). However, please note that the version may vary depending on your device's architecture (e.g., armeabi-v7a or x86). Google Play Store Apk Download For Android 5.1 1 Latest