Yes, I don't think Ilam is on access land. It's in a National Nature Reserve and there was a programme to dedicate NNRs as access land that started in 2015, but there were stated exceptions to this based on terrain and conservation, either of which might plausibly apply in this case.
https://magic.defra.gov.uk/magicmap.aspx is the most definitive online record and it doesn't show it as being access land.