Yes. Horses have a third eyelid, otherwise known as a nictating membrane. It is pink in color and slides from the inside to the outside corner.
It's visible as a pink rim at the inside corner of the eye. If it's more visible, it's often a sign of dehydration, infection (often of the eye - the third eyelid will try to protect the eye), cancer, a neurological problem or tetanus. Otherwise, the third eyelid extends up only when needed, normally when the horse is in a position where its eyes are in danger. Rarely, a horse might have a congenital problem fully retracting the third eyelid - in the one case I know, the animal ended up having recurring eye infections in that eye, probably because the third eyelid was not doing its job, and is now blind in that eye as a result.