I have a 2015 Mini Cooper Base F56 with BDC. The right rear turn signal stays on all the time with the key off until I pull the bulb or the battery goes dead. I have tried just about everything that I can think. I swapped the Left bulb holder with the right bulb holder and the right turn signal still stays on with the key off. I have left the battery disconnected for 24+ hours, the right rear turn signal stayed off for 24 hours until I drove it and stayed on all the time. I checked fuses, I pulled the BDC out and checked for any water damage, I checked the grounds in the back right, the connector is not burnt. I used BimmerLink to get the codes and reset the short circuit for the right rear turn signal. I called a Mini dealership, no recalls, TSBs, they have not heard of this as a common problem. I contacted ECU Pros and found out from the VIN that I have a Body Domain Controller not a FRM or FEM. I am not sure what else to try. I was going to run a wire from the front right turn signal to use as a trigger for a relay, to control the right rear turn signal, using the right rear turn signal wire as the battery supply for the relay in hopes this will trick the BDC to believe there is a bulb to prevent the warning during bulb check.

 

No surprise that you have a BDC not an FRM or FEM -- BDC is standard on a 3rd gen F series MINI.

+12v to that bulb comes down a wire direct from the BDC. There is no intermediate fuse, relay, etc. So one of two things is going on:

(a) your BDC is constantly sending +12v to that bulb (indicating a BDC problem)
(b) some short somewhere is getting +12v onto that wire from not-the-BDC.

This is the first I've heard of this particular symptom, and while my first thought would have been water intrusion into the BDC, you've already checked for that and ruled it out. And with the BDC being a computer module, I really can't imagine it just deciding to light that wire up all the time outside of a defect or tough to find failure like a solder joint gone bad.

So my money is on +12v getting onto that wire from somewhere else, which would be a physical damage scenario. In theory - big theory - one might test this by unplugging/removing the BDC end of this particular wire from the BDC. One could identify the connector that the wire is in, and see if the bulb stays on. If it does, that rules out the BDC as problem pretty quickly. If the bulb goes out, the connector to the BDC itself (and all wires within) remain suspect; one could hypothetically narrow things tightly by removing the wire running from the BDC to the bulb from the BDC connector (meaning getting that wire's terminal out of the connector so that it can be put back, not cutting the wire altogether). With that wire disconnected from the BDC, but everything else still normally connected to the BDC, if there is +12v on the wire it is absolutely coming from something not-BDC and the implication is literally an electrical short (something crossed with that wire).

I know it's not much to go on, but at the end of the day a bulb needs two things to light up: a wire with power and a wire to ground. You obviously have ground, and you're supposed to have ground, and that is present at all times. It's the power wire that is supposed to be switched and the question is whether the switch is failing (BDC), or that wire is getting power from an unplanned/not-normal place that the switch doesn't control (short circuit).

I will say that the battery disconnect for 24+ hours making a difference, and then the problem coming back 24 hours after that, sounds like a computer logic-involved thing -- and that would point to the BDC, because short circuits don't make timeframe decisions (they can't, they don't have logic of any kind). But I am hoping that is not your issue.

Hope this helps.