Many years ago now we lived in a very old house with a tiny garden. Like your house, it was surrounded by farmland and I was friends with the daughter-in-law of the dairy farmer who farmed the land.
Casually one day I said to her, if your f-i-l ever considers selling some land, we'd be up for buying it.
Fast forward a few years and due to the EU regulations it was no longer viable for him to operate as a dairy farm with the cost of the upgrade to his milking parlour. So before he put the farm on the market we got a message through my friend that he would be up for a discussion about us buying a bit of land. For comparison purposes, the bit we were interested in was only part of a large field which adjoined our boundary, about a quarter of an acre.
I consulted my brother (pretty savvy on all matters financial) and he suggested we should be prepared to pay £20k, as it was worth a lot of money to us to extend our garden. From memory this all happened about twenty years ago.
Anyway, we had a lovely, but very surreal evening with the farmer and his lovely wife. Copious amounts of gin and tonic were imbibed by us all, every topic under the sun discussed, apart from the crux of the matter of the land. Finally as it was obviously time to call it a night, the farmer said he'd sell us the land, he wanted £10k for it plus we cover all solicitors costs (both sides) and be responsible for stock proof fencing. We were happy with this and this is what transpired. This is the traditional old country way of doing things - and their family had farmed their land for generations.
I would add that if the land you're thinking of buying is adjoining your house, you might need to apply for change of use. We certainly did.
We're some of these terrible old people that you don't approve of, but needed to move near our daughter and downshift, so sold to some people from the Home Counties who snapped up our cottage - most of the quarter of an acre we turned into a lovely orchard. They were thrilled with it, particularly the lovely lady buyer - who said it was always her dream to have a country cottage. It made it easier to part with it to someone who really loved it.
Maybe get into a pleasant negotiating mode and off your high horse and you might be as fortunate as we were.