There currently is one option to travel by night train from Copenhagen (Denmark) to Hamburg (Germany), taking approximately 6 hours: Namely the EC 1195 operated by DB.
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. These trains only have first and second class seats. The cars remain lit throughout the night and the seats cannot be reclined.