Yes, it’s called the Duty Recorder function. You can record and save up to three recordings, but only in everyone in the party consents, and it disables battle logs so you can’t actively look at outgoing damage from players. Like I mentioned before, I’m personally of the opinion that it’s a waste because there are already third-party streaming and recording sites/features that aren’t so incredibly limited. You can also only view the recordings in an Inn Room, so they can’t be used real-time or after a wipe for immediate analysis.