According to Alfred Hedges' book "Inns and Inn Signs of Norfolk and Suffolk, the Ram stood where the Rutland Arms is now situated. The Ram itself apparently replaced an even earlier inn whose name is unknown.

The Newmarket Pubs website tells us:

There was an Inn here earlier but the name has never been found. The Inn was renamed after the Earl of Offord (an eccentric who had his coach drawn by stags) was chased by a pack of hounds and sought refuge in the yard of the Inn. The word "ram" being associated with a strong scent, the Inn was re-named The Ram.