No, it's part of the Peak District national park (as are Stanage, Burgage, Padley, etc.) and has been designated an SSSI. As such it's covered by various by-laws, the relevant one being that no vehicles (inlcuding bicycles) are allowed on there, in order to try and preserve the natural landscape etc. and not have it churned up by tyres and the like.
The park wardens patrol around and kick anyone in breach of these by-laws off. I had a long chat with one of them once and he said that he was aware that trials riders, who stick mostly to the rocks, don't churn up the soil etc., but couldn't let us on as it would set a precedent for other types of cyclist, motor bike riders, etc. He was a pretty nice guy, but another warden who kicked me off was much less friendly and even threatened to call the police when I asked what he wanted to take my name for (they take your details and sometimes photograph your bike and if you're caught up there more than a couple of times they fine you. Nothing a fake name and address can't sort out though ).
Generally though, if you pick your time and spot well, you can avoid all this