After many routine lowers services, I undid the nut about three or four turns and gave it a firm bash, but the nut wouldn’t fully unscrew – it just stayed on the very end of the thread, all wobbly like.

When it eventually did come free, I had the nut, a small silver washer and a very thin black curved plastic washer, which I don’t recall seeing before. Something tells me this plastic bit coming away from the lower or shaft was the problem. Do I need to order any new parts do you think?

The sliver of metal is likely to be a small piece of thread left over from the machining process.
The plastic washer is a crush washer the same as a standard one found on other rockshox forks, it sits in the recess on the rebound bolt, replace with a new one when reassembling
HTH. 😀