LOTS of good lanes on Exmoor.
Some with a rocky base that can be driven in pretty much any weather, some a bit more muddy and best avoided when really wet, been a few years since I was there and I'm not sure whether the NERC Bill has left any driveable ones :(
Do a search for Tarr Steps, Hawridge Ridge and.....errrr
Someone else with a better memory than I now have will help you out more