Sure, this could easily be done as what's being displayed is just a video feed. You'd just need to leach off of the feed, which would then need to be provided to a recorder. That could be spinning disk, SSD, thumb drive, or flash card. That all depends on they type of recording device being used.
There are systems out there which do this, but none which are manufacturer installed, as far as I know. The primary purpose of the backup camera is to be used when the vehicle is backing up. It isn't on at any other time. This seems logical at the outset. In order for it to work full time, you'd have to have the camera setup or changed to be able to handle the situation. OEM cameras aren't made to do that from the factory.
As stated, in order for this to work, you'd need the camera on all the time. You'd need it to be recording all the time. That takes up a lot of memory space, depending on the resolution which you're recording at. There are also ways you can make it so it would only come on when there is motion sensed around the vehicle. This seems like a more attractive alternative, but you'd need some way to sense the motion. "Always on" systems would also be a constant power drain on the battery while the vehicle is shut off, so another something to think about.
While this is doable, it would take some engineering to make happen. I'm sure there are systems on the market today which can provide such a convenience, but you'd have to do your own shopping to find one.