Michael replaced the socket for me - thanks, Michael!
However, after doing so, the situation appeared slightly worse than before - the unit seemed totally dead in the car. Worked on AC, though. A bit of (very awkward) prodding revealed that the ignition sense line was unconnected at the car. Mike's player worked okay, but we suspect that it may have ignition sense tied high internally, as it doesn't shut off properly.
As a workaround, I've been running for the last week with a spare power jack in the socket, and the player in "Force DC" (thank you, Mark!).
I took it back to Cambridge Car Audio today, and they sorted it out - both ignition sense and dimmer were unconnected, and both work perfectly now.