Peugeot 806 MAF sensor issues

Sounds like a plausible diagnosis. A faulty MAF sensor can definitely cause those symptoms. Has the check engine light been diagnosed, or did you just read the code? Have you tried cleaning the MAF sensor with a MAF sensor cleaner? Sometimes that helps, but if it's truly gone bad, cleaning might not do the trick.

I only read the code. I haven't tried cleaning the sensor yet. Where is the MAF sensor located, exactly? And is it difficult to remove and clean? Also, if the sensor is bad, can it damage other components?

The MAF sensor is typically located in the air intake duct, between the air filter box and the throttle body. It's usually held in place by a couple of screws or clips. Cleaning it isn't too difficult, but be careful not to damage the delicate wires inside. As for damaging other components, a faulty MAF can cause the engine to run rich or lean, which could potentially damage the catalytic converter over time, but that's less common. Given the symptoms and mileage, it might be time for a replacement. Best to get it checked by a workshop.

Thanks for the advice. I took it to the workshop, and you were right – it was the MAF sensor. They replaced it, and the total cost was 365€. The fuel economy is back to normal, the engine isn't stalling anymore, and the smoke is gone. Appreciate the help!