I like your thinking! But this would mean (since the functionality of the switch was tested) that putting the vehicle in reverse would turn off the lamps and return the mirror to home.
I'll try it anyway! Stand by for result...

Well not necessarily, the switch may complete the circuit all the time and break the circuit to turn the lights on. Remember these switches aren't hardwired to the bulbs they are connected to a control module so anything is possible.

Who knows but it's worth trying every avenue in order to save money......

Looks like you only disconnected the sensor, however you can bypass the sensor to test if it's the sensor or the wiring (maybe a short).

You can bridge the terminals from the harness to manually switch the rev. lights on and off and this checks the wiring.

I've attached a picture for a Honda and instructions (hopefully you can follow) having the same issue.

Since you are driving an MT, there is an easy way to test the switch.

Yes the key has to be on (engine off), otherwise the lights would still be on when u leave it parked in rev. gear and take the key out and walk away....

Put it in neutral and turn the key to "on"...your rev. lights will be on, if ur having the problem you described,

Now unplug the sensor from harness. The lights should no longer be on. You can bridge the terminals (SEE PICTURE-Hopefully a similar 2 wire setup for the X5) from the harness to manually switch the rev. lights on and off and this checks the wiring. If they didn't turn off at all, make absolutly sure u unplugged the right thing and then you can check the wiring.

QUESTION: Just to be clear, when you unplugged the sensor did the lights stay on? If they did then you are dealing with a "short".

So what's the verdict?

Thanks for the ideas, but in the original post I mentioned that the lamps are still on even with the sensor disconnected. I checked the harness for a short by testing for continuity between the two sides of the harness plug, but the resistance was still infinite (no short or contacting wires).
The sensor/switch works fine, as tested with a multi-meter.
As suggested I bridges the sides of the harness plug, but no change.

I reckon you have a short. It is electrically impossible for the light to be powered on but have no voltage coming through the harness.

You have a short somewhere so you'll have to find it

To be honest if it were mine I'd pay the $10 for a new switch, your multimeter could be bad or you might be doing it wrong just seems easier to me to get the switch and then once you've ruled that out you can focus on running up the line from the harness to find the short.

One question I didn't ask is how long have you had the X and has it been like that since you got it? Some people retrofit a switch into the reverse light so that they can turn it on for hooking up a trailer in the dark etc

A new switch was the first thing I did!
Besides checking with a multi-meter, I've traced the wires from the switch to where they disappear into the dash with a huge bundle of other wires with no luck.
It IS possible for the lights to be on with no current in the harness: in spite of the physical switch being similar to old ones the circuit is different. In old style circuits, the reverse light switch controls the current that goes directly to the lights (as simple a circuit as you can imagine); in newer Bimmers it just sends a signal to one of the modules to tell the computer the vehicle is in reverse (so it tilts the passenger mirror etc). If there's a problem with the module, the wiring might be fine but the car still thinks its in reverse.

Ahh we are talking about different harnesses obviously - I'm talking about the one that plugs into the bulbs at the back of the vehicle.

Good luck with it

With the lighting control module (LCM) taking sensor inputs and driving outputs, it appears to be your issue. Just as you have checked for shorts at the switch, you could check the plugs on the LCM to see if there is corrosion there or visible damage. I don't know how many separate lighting modules there are, but the schematics are available.

Just curious, was the instrument cluster changed?

The "Reverse detection" function is only active when this function is correspondingly coded in the instrument cluster. If the instrument cluster is coded for manual transmission, the program and selector lever display for automatic transmission is masked out.
The information "reverse engaged" is made available in the form of a telegram via the body bus (K-bus ).