Continuous beep with key turned in ignition

I have a UK spec. late 2003 BMW 325i Sport saloon [E46]. It's a petrol engined car with a manual box. It's been very well looked after from brand new and has led a very easy life.

Having not driven the car for a fortnight, I went to start the car last week. When I placed the key in the ignition and turned to position 1, the mini speaker [or maybe its piezoelectric] beneath the instrument binnacle emitted a constant beeping noise. The audio frequency of the noise is similar to the reverse parking sensor when you're very close to an object, or the blip the temperature alarm emits when it crosses the 3C threshold. The beeping is continuous whether the engine's on or off and whether the car's driving or not.

I've had a good look around various forums, including this one, and other people appear to have had similar problems, but unfortunately those threads haven't returned solutions. Ideas could be :

o A fault ambient temperature sensor, however the temperature gauge reading on the dashboard is accurate.

o Faulty rear parking sensor(s), however I believe that these are trapped by fuse number 24 in the fuse box and having removed that fuse, the sound is still there.

o Faulty parking brake sensor, however I believe that would not usually emit an audio signal when the car is stationery.

There are no warning lights illuminated on the dashboard... so I'm really at a loss to understand what it could be and would be extremely grateful for anyone who has any ideas. Thanks so much.