In the unlikely event that Sony one day shuts down the PSN for the PlayStation 4 console, players who would run out of battery in the console would no longer be able to play even the games they have on Blu-ray media.
Fortunately, this has now been corrected with a new version 9.00 system update. As long as the console is updated to that version, games run without crashing even when the battery is not functional, i.e. when the console is not connected to official servers. Supposedly this is true for both physical and digital games. It is also possible to win trophies in games, only the time of winning the trophy is not recorded.
Sony has not officially announced this repair, and this discovery is based only on early testing. The problem seems to be solved, but the person who first pointed out this potential problem plans to do more testing to be completely sure that gaming on the PS4 console will not go extinct in the event of some global apocalypse.