Thanks for all the suggestions, which I appreciate.
It was a software problem: in the end I did a factory reset and that cured it.
I hadn't wanted to try that until I'd exhausted all the other possibilities, because if it had been a hardware issue (damaged antenna, for example) I would then have had an unregistered Kindle that I wouldn't be able to register - not without WiFi. And although I normally use my various devices in aeroplane mode, there are some Kindle features that you don't get if the device isn't registered.
Thanks again
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