Although not obliged to, and as mentioned above, PRoW departments are mostly responsible for paths on the definitive map. However, permissive paths are still an important part of every authorities network, mostly due to 'missing links' on promoted routes (AFAIK something like 70% of the Pembrokeshire Coastal Path is along permissive routes), and myself am keen on permissive routes through farm yards (assuming the definitive line is open) and waymark them with white 'permissive path' waymark discs.
As such, most PRoW departments will keep an ad-hoc record of permissive paths (I always forget to add them to our mapping system TBH), and as they aren't recorded on any map - as they don't have to be - you'll need to rely on randomly finding one, or just hoping that PRoW departments can find that magic five spare minutes and get them added to their own website mapping system (which they should all have BTW, with online definitive maps).
I'd rely on the defra map linked above rather than ringing your local PRoW department as they'll likely refer you to it anyway. I've not actually seen the defra map before, it's much better than the
welsh one which has just reminded me I've not updated for the last 3 years
although the routes shown are ones created as a result of agri-environmental schemes (e.g. they get paid for it), but not that it makes any difference really.
Clicky if you'd like to know more about permissive paths.
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I've never come across a permissive path that the landowner wanted to open up without getting paid / given something for or not as part of a longer route. Chances are you're really so you're not missing out on much