Try downloading another game, preferably a free-to-play one, and Dying Light simultaneously. It often helps if two games are being downloaded at the same time.
Also, have you got an external hard-drive added to your console? If so, uninstall the game from the external hard-drive, disconnect the hard-drive and install the game in a default manner.
Additionally, perform the steps presented in the following article: Black Screen - Clearing Cache.