Yes, PlayStation 3, 4 and 5 have all need additional space while installing the game. I always thought this was silly then discovered that if you play a game on Xbox while an update is downloading, the download crawls.
I found this out the hard way playing Final Fantasy XII, and started playing as soon the first game chunk had downloaded, and thinking the download rate for the rest of the game would be constant, was surprised when I hit a wall beyond which you can't pass until the full game has downloaded. Close the game, download speeds up, start the game, download slows down. ¯\_(ツ)_/¯