I was playing fine on Medium settings, with average FPS around 80 on my 7700k and 1080 Ti. But after visiting my appartment and sleeping for the first time I woke up to be stuck around 20 FPS, sometime 5 FPS...

Has anyone else had this?

Not yet, haven't gotten that far. But they did release a new Nvidia driver specifically for Cyberpunk today-460.79. Perhaps try the new driver and see if it helps.

I don't have the game, but try to use MSI Afterburner to check your resources in game, maybe the game is not dumping memory and that cause the fps decrease with the time!! If you haven't update your drivers yet, do that. If you need something else, let me know and i will try to help you!!
I don't recommend you to open a ticket because the system is overwhelmed, it will take days to reach you!! Wish you luck!!

I tried setting the graphics settings to low, and the FPS dropped to 10.

When I'm in the menus its 144, but when I resume the game it drops down from 100 to around 10-12 over about 5 seconds.

Totally unplayable now.

Yes, that it is, you have the same problem witj other games? Looks like the driver is the problem, if you have the latest one try to unistall and install the older one, if you haven't update the drivers, Nvidia and AMD release some new ones for the game yesterday. Peobavly that will solve your issue.

Have you checked the temperature? Might be clocking down due to heat.

Temps were a cool 30 degrees c (water cooled)

I disabled my overclock (which was a subtle one anyway), rebooted and the frames have increased to 65 FPS. They're a bit lower than the previous frame rate, probably due to removing the overclock.

I didn't have this issue in any other game prior to this evening, but then I did just install the Game Ready driver before playing Cyberpunk.

I think I'll put up with it, as I'm going to be changing from my 1080 TI to a Radeon RX 6900 XT this weekend, hopefully this issue will be a thing of the past.