A couple of years ago an Irish racehorse failed a dope test because there were traces of nicotine in its sample. The trainer expressed surprise.It had finished second. The next time it ran without any trace of nicotine being found.This time it won. The trainer, in Irish manner, said that he knew it'd be a good horse once he'd got it off the cigarettes !
So the question is: In what way could nicotine affect a horse's performance ? (It may be that it is not banned because it adds stamina or speed but because it has some other effect, for example that it masks other drugs or it eases pain in the horse's joints in some way. Certain drugs that ease pain are forbidden as being unkind to the animal).
I don't know whether nicotine in particular has been found to have a significant effect or if, since it's a horse, they're being cautious with something that would seem to have no legitimate reason for being there.