HMS Zebra was the first ship to bear the name in the British Navy. She was a 14 gun ship sloop of the Swan class, launched on 8 April 1777.
Lieutenant Sir John Orde of HMS Eagle was appointed Captain in 1777 after her Captain, Henry Colins died the night of his arrival at New York, 25 September, with its convoy.National Archives (United Kingdom) ADM1/488/2 ff. 3-4
She was abandoned and blown up after going aground on 22 October 1778 at Little Egg Harbor, New Jersey, during the American Revolutionary War after only one year in service.
References
Winfield, Rif, British Warships in the Age of Sail 1714-1792: Design, Construction, Careers and Fates. Seaforth Publishing, 2007. .