This is a long shot maybe.

I had a dead tailgate release button and one dead licence plate light. The other light was working but very dim.

I investigated and found that a seal had perished badly and was letting water into the switch and licence plate lights. I replaced the seal, cleaned the corrosion off the light contacts and dried out the switch with a hair dryer. Hey presto, both lights back on full strength and no problems from the release button any more.

The seal in question. I don't know what you call it but there's this "strip" of trim above the licence plate area, attached to the very bottom of the tailgate. For example if you were opening the tailgate using the release button your thumb would be on this piece of trim. That can be removed by undoing three or four torx screws on the bottom of it and then the seal in question is visible between the strip and the bodywork.

Mine had almost turned to dust it was that perished !

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2001 X5 4.4i