As Gordon said, find your local ALRC Land Rover club.
I pay £20 per year to be a member of mine. They organise regular events including green laning. Membership of one ALRC club allows you to attend and participate in any event organised by any other ALRC club around the country just by showing your membership card. Each club is different in the events they organise - some are competitive and some aren't. Mine is competitive which means that the majority of the events are trials based on navigating round a course of numbered cane gates. TYRO and RTV trials are designed to be non-damaging and are for standard vehicles. CCTV trials are the proper hardcore ones that require roll cages etc. Others organise pay and play days.
The clubs will also have someone in the organisation that will mark local lanes on your OS map for a small fee - our guy charges £3.
Also, your local council will have information about what lanes you can and can't use but some make it harder than others to find. I think it's called the definitive map.
I've been greenlaning in my P38 and not suffered any scratches or damage but it will depend on the lane and weather etc.
DevonP38, could you PM the website you mentioned? I'd love to know more of the lanes around where I live.
2001 P38 4.6 Vogue
2019 Mazda MX-5 1.5
2016 Honda Civic 1.6 diseasel