There currently is one option to travel by night train from Odense (Denmark) to Berlin (Germany), taking approximately 7 hours: Namely the RJ 387. These lines are operated by Deutsche Bahn and DSB.
Deutsche Bahn (DB) exited the actual night train business in 2016. However, it operates around a dozen Intercity and Eurocity trains that run exclusively at night.
In these night trains, there are only first and second class seats. The lights stay on all the time and there is no way to recline the seats.
DSB (Danske Statsbaner) is Denmark's national railway company, operating passenger trains across Denmark as well as connections to neighboring countries.