When the ignition key is turned off, the three brake lights and instrument panel lights turn on and stay on. To get by I remove 5A fuse no 43 which (I think) cuts power to the panel. I then lock the car at the door.
A possible clue is that disconnecting the battery overnight and reconnecting the next day, can almost make the problem go away. But a while later the alarm sounds and lights flash. If I clear this by starting the car, turning it off will recreate the same "lights on key off" problem. So I'm now using the key without buttons to lock the car, just in case the problem relates to the alarm system. Both of the two button keys work similarly.
Two nice local garage suggested it's an earth fault; or water's got into the electrics but they say it'll take time to investigate. I can ill afford this so please someone shout an answer and I'll report on its success!
The car played the same silly game two winters ago when the battery drained down. Disconnecting the battery overnight and fitting a new battery (after 8 eight years) made the problem go away, till now. Thanks for help you offer.