If this person has been paying the training and board bills on her all of this time, "Yes!", they can claim they have a lien on her or the barn can claim they have a lien on her. In which case you would have to prove that you did not agree to keep paying the bills. In court, they would have to go to prove that you agreed to continue paying the training and board on the horse.
This is a 'civil matter' and not a 'criminal matter' if they have been paying all of the bills.
In this situation, "Possession is 9/10 of the law".
I would NOT call the sheriff or he will tell you to go to court and to leave the horse exactly where he is. (See ^^^) Just go unannounced and pick up your horse. Then THEY have to prove that you owe them for the training and board.
If they go to court and win, the court will order you to give the papers or pay all of the training and board bills.