Radio & Clock stay on after car is turned off

[pmcinema]

I have a very clean 1991 318is. Just recently, the non-stock radio and clock stay on even after I have turned off the engine and removed the key. Both items have turned off with the turn of the key for the past 4-5years. Since they are staying on, subsequently my battery died and had to be replaced this past weekend. I have checked the fuse box, all seems ok. Where else should I check?