Airplay or Mirroring without WIFI or internet connection??

Hi. I assume you have Apple TV 4 or 4K running tvOS 11.1.

Go to Settings > Airplay and allow "Everyone".

Enable WiFi and Bluetooth on your iPhone. WiFi must be turned on, but doesn't need to connect to a network.

Go to enable mirroring or AirPlay in Control Center on the iPhone.

Wait a moment for the Bluetooth scan.

Apple TV should appear, and you'll be able to connect.

If you have a 3rd gen (REVA version) or newer Apple TV then you can use peer to peer airplay. However, content like iTunes would require an internet connection

No problem. Are you sure you have the 1st gen (big aluminum box with hard drive) as then you wouldn't have access to airplay at all.