For two weeks now I try to solve the problem, but still without success.

21 hours ago, ungross said:

what is the probability of getting the required file if I use the 3rd option?

Not very likely, you can try or ask for help on the btrfs mailing list or on IRC #btrfs

What should I do?

If the recovery options on the FAQ don't work I can't offer further help, like mentioned you can ask for help on the btrfs mailing list or #btrfs on IRC, and this won't help you now but for the future you should always have a backup of any important data.

I will now with this command:

btrfs check --repair /dev/nvme0n1

try to fix the pool.
What exactly happens there?

Is maintenance mode required for this?

I've never heard of backing up the data from the cache. That's new to me.
Are these not written regularly to the array?

9 minutes ago, ungross said:

btrfs check --repair /dev/nvme0n1

nvme0n1p1

 

9 minutes ago, ungross said:

Is maintenance mode required for this?

Not if it's not mounting.

 

6 minutes ago, ungross said:

I've never heard of backing up the data from the cache. That's new to me.
Are these not written regularly to the array?

You can have cache data moved to the array on a regular basis, depends on how the share is configured, but you will likely have some data residing permanently on the cache, like vdisks, and that needs to be backed up, unless you don't care if it's lost.