DTC, thank you. I have just mailed and hope it is not bounced back as my first effort was with another so-called CEO (dating from 2008, so presumably out of date.)
I've just found an email from Barclays showing the sending server as mta.emails.barclays.co.uk, with an IP address of 13.111.88.80. So that's something else that you might see within the extended headers. (NB: Always check the 'From' field that's got lots of numbers in it. The one that shows what appears in the 'From' field when you view the email normally can easily be faked).
There should be an 'Unsubscribe' option at the very bottom of any emails from Barclays but DON'T click on it unless you've checked the extended headers first, in order to confirm that the email is genuine. (Click on 'Unsubscribe' in a dodgy email just tells the spammer/scammer that they've found a genuine address to send their rubbish out to, vastly increasing the chances of you getting far more from them!)
In Thunderbird, look for the line with buttons labelled 'Reply', 'Forward', 'Archive', etc. The right-hand button is labelled 'More'. Click on it and select 'View Source'. It's then the second line down that you need to be looking at.
Buen; thanks for your help. I've found what you suggested, but it means nothing to me. Anyway, they tell me they are actually going to sort the problem.
Bobb I did view source and couldn't understand it, although there did seem to be references to Barclays, and no Japanese or weird character stuff that I could see.