I have a collection of17 rescues at the rescue I co-run, each of them are either 'untrainable' or seriously over trained (broken). For the longest time I truly thought the ones that were labelled 'untrainable' really were.
There was an 18 year old pony we got at our rescue who had lived her life completely wild on someone's 20 acre property. It took 6 men 5 hours of herding to get her into a stock trailer to get her to us. When we got her we worked very hard to get her to the point where we could lead her in and out to her paddock - but that was enough. As an 18y/o terrified animal we decided it was best for everyone to just let her live it out.
A little girl volunteer at our rescue saw something more in our wild pony. She spent 2 weeks sitting in front of the pony's stalls, then in the stall, until the pony became curious. Eventually she could pet her whole front half without issue. When she went into the paddock her pony would follow her every where, even over poles and little jumps (At liberty!). We thought 'that's so sweet, but she can't actually learn anything useful', because you can't take the typical approach with her. Most horses you teach them to yield to pressure. With this pony if you look at any part of her squarely she's yielding as far from you as she can get. So we let her little girl play with her however she liked.
At this point I started venturing into clicker training with one of our 'untrainable' ponies who had been a stud too long. He had violently attacked his previous NH trainer. I got the pony to start targetting the end of a crop and a foot ball and backing up on a verbal cue. He was doing great - I explained the process to the girl who loved our wild pony. She started. The pony quicky picked up the concept. She recently had her first un-tranqued farrier session because she had become so comfortable having her feet handled by the little girl - with clicker training. More and more this pony is learning, now her little girl can lean over her back resting and hug her around the top. The pony is too small to ride, but she has done SO well! She will target things to jump over jumps and go through obstacle courses with 'scary' objects. I'm amazed!
My pony, the violent one, has found a better way to let out his energy - fetching the foot ball for my fiance :P They love playing together now!
The little girl and her wild pony 'jumping' :P