What's your account type - POP or IMAP?

I am connected to pop.gmail.com , and I have selected the options "Leave messages on server - until I delete them". FYI: I detatch the files with the right-click option "Detatch All" and put them in an appropriate folder on my drive.

I'm speculating a bit here, but this may be what happens. When detaching an attachment, the original message is saved again without the attachment, and the original is then deleted. The Gmail server seems to interpret that as 'until I delete them'. What's left is the local copy on your computer.

When multiple devices are used to access an account, IMAP is the more elegant way. You wouldn't have that behavior with IMAP. Verify if IMAP is an option for you. https://support.mozilla.org/kb/imap-synchronization

Thanks for the idea about IMAP. I have now tried it (set up a new account in Thunderbird as IMAP with all the settings and disabled the old POP account) and it does exactly the same thing. As soon as I "Detach All" it then deletes it from the gmail server.

This isn't the way IMAP works. If the message would have been deleted on the server, you wouldn't see it in Thunderbird anymore either. What's left is the message with the attachment stripped off. The whole idea of IMAP is to manage your messages on the server.