Finally it turned out to be a network problem. AppleTV indicated a local IP adress. Restart of a WLAN repeater (AVM Repeater with RJ45) connected to the RJ45 of the appleTV solved the issue. Airplay works fine with appleTV 3rd Gen.
A bit strange is the effect that appleTV despite loosing IP connection did show a reaction to airplaying something - hourglass and then and error message (cannot load .. retry later). But I learned that appleTV does also have a WLAN on board, but that would be unused if connected to RJ45 network. I couldn't even find the WLAN settings in the configuration menu. Also the Box has Bluetooth - at least for remote control and keyboard. Apple states that WLAN is unused when using RJ45 network, but appleTV seems to appear in the WLAN Router list as WLAN connection .. however I don't need to understand everything if it works again ..