Kind of looks like a cordless phone battery IMO.
Yes it's actually very very close to many Nokia mobile phone batteries I found now. I've ordered one off Amazon to see if it works but it's a little bit different size ... may be a fraction too big. It turns out to be similar in spec to one used in the Logitech Harmony remotes but that one again is a little thicker. But as least solid options/alternatives coming up thanks to support from Audiokarma members
This Nokia one I found is a better fit and will be a plan B. But I don't 100% know the wire to metal plate connection order .... red / black etc.
https://smile.amazon.co.uk/Nokia-re...s=3.7v+mah+lion&sr=8-14&tag=amz-mkt-chr-uk-21