Well I am a believer in coincidence so had wondered if there was some local issue.
However it is a strange coincidence that these screenshots are from two separate PC's, both had been upgraded from 3.7.1 (to 3.8 IIRC) then to 3.8.1 and finally 3.8.2 and for some reason neither PC will update any extensions now yet they did when the install was 3.7.1. there is ample free space available on both.
There is only one commonality, that of how the upgrade was performed. Each upgrade was accomplished by downloading and mounting the new ISO, copying both bzimage & tinycore.gz from the mounted ISO to the boot directories effectively replacing the old with the new then rebooting..
Ok so I fired up an old VM with tc v 3.7.1 on it, checked the tce directory and it's sub-directories, all have 775 permissions and set to tc user and staff group, this system also will no longer update any extensions however did a week ago when it was last used. ..
interestingly as a user I have full control to copy, delete any files in and out of the tce directory. I downloaded a movie directly to the tce directory and played it then copied it to the optional directory and then deleted it, all as a user only.
This would indicate the issue is neither one of permissions or space..