There currently is one option to travel by night train from Berlin (Germany) to Innsbruck (Austria), taking between approximately 9 and 10 hours: Namely the UEX 1305 operated by UEX.
UEX (Urlaubs-Express) operates seasonal night trains, including car-carrying trains (Autoreisezüge), from Germany to Austria and back.
You can choose a seat, a couchette berth, a whole couchette compartment, a private sleeper compartment or connected sleeper cabins.
Cars can be taken on the train at the car terminals only, and the vehicle is booked together with the ticket.
Skis travel free in the ski compartment. A bistro on board sells drinks and snacks.
Children with their own seat or bed pay a reduced fare.