Thank you for your support so far.
I ordered a fan and will give an update after being installed.
It might be of interest what I noticed prior to detecting that the fan is not working.
The Mini was upgraded from 4 to 16 GB two weeks ago, knowing that 8 GB are supported by Apple.
The upgrade went through without any problems.
The difference in speed was quite remarkable. But after running e.g. a video in HD format and then opening a digital darkroom app the Mini became very slow; two times the Mini shut down without notice. It took me a while to detect that the fan was causing the problem.
So, hopefully the high temperature has caused no damage.
The Mini is running so far but now waiting for the fan.