Sounds like the vehicle is not fully turning OFF, it's in the accessory mode, which can be a faulty electronic (BDM) module. All faults may be related.

 

It seems now that it's totally random if the the door open/close symbol on central display and the radio/nav screen switches off when I leave the car. Last night both (and the mentioned button) stayed on for at least an hours after I left the car. This morning the screen had switched of, but the display was still on. The button is always lighted. Sigh...

 

I accidentally found that after turning off the ignition, if I opened the glove compartment, the display, screen and switch light turn off. When I tilt the lid up and down a little, the display, the screen and the switch wake up and go off. Must be a loose wire or similar near the glove box?

 

To end this post: I found that when lowering the glove compartment it pushes against a Defa interior heater mounted on the center console wall. When I push on the heater with my hand, I can make the display, screen and switch light go on and off. Seems like the one installing this heater just screwed one or several screws into wires behind the wall.