Today, Patch 1.02 (equivalent to 1.0.5 on PC) released for PS4 versions of Enter the Gungeon. There were unforeseen consequences. Almost immediately after the patch was certified and pushed live, some issues were identified that caused save file loading failure for certain players. In response, the patch was rolled back to 1.01 (equivalent to 1.0.0 on PC).
Unfortunately, this rollback is not happening automatically for certain players. For an immediate fix, players can delete and reinstall the game. By reinstalling, you will return back to 1.01, and resume your saved game. Any progress spent on the new patch will not roll back to 1.01.
According to co-founder and programmer at Dodge Roll, David Rubel, the issue is being worked on with Sony. There's no definite news yet, but as more information is released and possible fixes are issued, this post will be updated.