I’m not familiar with the iPod classic however you problem seems to be a common one.
From what I could find online this is a common problem people come across with their iPod classics after using them for some time.
It seems that after sufficient use, the click wheel will simply stop working (in your case, just the play button.) other people have had the entire click wheel stop responding at once.
The only solution I have been able to find requires the replacement of the click wheel.
Luckily it is fairly easy to find replacement click wheels for iPod classics’. Replacement parts are avalibe here on ifixit (link below, with guides for replacement) but are currently out of stock, it shouldn’t be to hard to find them elsewhere should you not want to wait till the part is restocked.
Hope this helps!
Link to replacement scroll wheel: [product|IF130-007-2]
Guide links: [guide|568]
[guide|571]