Help! Ignition Chime keeps beeping when Key is out

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Hi all,

Car is a 2004 x5 3.0i.

Whenever I pull my key out of the ignition, the chime keeps beeping when the driver door is open. It's almost as if the car thinks the keys are still in the ignition. This is causing my keyless entry not to work.

I also had an ignition problem last month where the car wasnt starting and the cluster lights were all dim. I pulled my ignition switch out and sprayed it with contact cleaner and it solved that specific starting problem.

As for the beeping, I'm wondering if I should replace the ignition switch with a new one or does this sound like a problem with the ignition lock cylinder? I would really like to avoid changing my keys.

Any help would be greatly appreciated.