Why, yes, they are for sure there. I have seen them. What a treat!
Here's a little snippet about them:
Late 1400's: Columbus brought the first Iberian (Spanish) horse to the island of Hispaniola and started breeding ranches to supply the conquistadors with mounts for their explorations. The soldier-explorers usually rode stallions but Columbus made certain that mares were included whenever a ship was coming to the New World.
Early 1500s: Lucas Vasques de Ayllon attempted to settle the mid-Atlantic coast. Of the 500 colonist brought from Hispaniola, only 150 souls survived to catch a passing ship back home. The 89 to 100 horses brought by the colonist were abandoned when the colony failed.
Late 1500s: Sir Richard Grenville traded for supplies and animals with the Spanish in Hispaniola and brought horses up the Carolina coast to the English colony at Roanoake. His ship, the Tiger, grounded in an inlet through what is now know as North Core Banks, and had to be careened (rolled over on her side) for repairs. Livestock, including horses, would have either been cast overboard or released.
There are mentions in records spanning the years between then and now of these Banker horses. These descriptions of their size, toughness, temperment, etc., have varied little over the generations and could be easily be used to describe the Spanish stock from whence they came. There are oral histories and traditions handed down faithfully from one generation to another of the coastal people ..."The horses have always been here. They swam ashore off sinking ships".