Someone on Reddit linked to this announcement on FFXIV's Twitter page:

https://x.com/ff_xiv_en/status/1807117370919702637

FINAL FANTASY XIV
@FF_XIV_EN
We are receiving reports from some customers on the PlayStation®5 version where they are being disconnected from the game when the clock hit 12:00 a.m. on Sunday, June 30. When attempting to re-launch the game a countdown timer to the official launch of Dawntrail is shown which stops them from progressing.

This seems to only occur when certain conditions are met; however, we are also receiving reports that affected customers are able to resume playing by downloading the Free Trial version. If you experience this issue when the clock turns over to Sunday, June 30, please try downloading the Free Trial version.


In short, it recommends downloading the free trial version to get around the block, but I can't see that option on the PS5 shop search.