The situation has changed:

~> LC_ALL="en" touch test
touch: cannot touch `test': No space left on device

A proper file system check (e2fsck -f -D -C 0 /dev/disk/by-id/ata-WDC_WD5000AAKS-60A7B2_WD-WCASY3646118-part7) fixes the problem temporarily, meaning that I can then generate a few files, after which the problem pops up again.

I believe this is an ext4 file system problem.