Ok that is the reset procedure instead of a button. It'll do the same thing. Unfortunately, you may have to have it fixed/replaced. It sounds like as if its a mechanical issue.
As its an E39, you might want to think about an aftermarket CD player to put in the dash. That'll probably be cheaper than replacing yours. Unless you use your cassette, its a good way to go. Most of the newer ones have iPod controls, more power than the stock player, better AM/FM tuner, better CD player, etc.
Good luck with your decision.
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