I wonder if the car has a battery on the insturment cluster board the same as my '88 Cabrio did? If so, that could be the problem. Once the lithium battery fails to hold a charge it has to be replaced, and the service lights were the indication that it needed to be replaced. On the '88 it was a fairly easy DYI job if you can solder. I'm just not sure if the newer models still have the battery on the board or not.
1990 535i 5-sp., except for 16" M-Contours, cd deck, and clear corners, completely stock. 226k and running strong