I had a look around Amazon's Kindle forum, and other people had been having similar issues - all seemed to be fixed when they took the Kindle out of the cover, or with a cover with a light. Some people had had four or five replacement kindles first.
I dropped Amazon customer services an email, explained the problem, explained that it was a gift, as was the cover.
Within 30 seconds of pressing send, Amazon were on the phone. Very apologetic, had I tried reboot (yes), had I tried without cover (yes).
Well, sir. What we'll do is credit your account with the cost of one of the cases with a light (£50.99). No need to return the original cover as it's clearly defective. Sorry about that, hope that's ok?
Crikey. Absobloominlutely. Amazon Customer Services FTW!
So, now I need to decide what colour I want. The burgundy red (colour of the original) *is* rather sumptuous. But I quite like the burnt orange too...