I comletely disassembled and cleaned out the insides of the antenna motor box, greased it up with white lithium spray grease (Track Auto, remember those guys?), excersied the antenna mast and cleaned it as best I could, reassembled all (getting that toothed Teflon rope in correctly took a trick using fishing line tied around the first tooth and pulling it as the rope got stuck at the end of the lowest tube guiding it into the gears), and guess what - it worked!
If the mast starts sticking again, then I'll throw it into the garbage heap, but for now everyting looks great.
Thanks to all of you for the advice!
1989 560 SEL (black/black)
2001 Audi TT Roadster (silver/grey)