A starship named Enterprise was in service sometime in the 32nd century. 

Ambassador Spock’s efforts to unite the Vulcan and Romulan people 800 years earlier were finally fulfilled and his legacy lived on through archived holographic recordings.

Ship Model (None)

As mentioned in Star Trek: Discovery - “Stormy Weather” (TV, 2021)

In the fourth-season episode of Star Trek: Discovery, “Stormy Weather,” Saru gave a status update to Captain Burnham about Starfleet investigating a subspace rift left behind by the Dark Matter Anomaly (DMA). Saru casually mentioned that, on encountering subspace rifts, "Enterprise noted heated plasma." This could, of course, refer to past iterations of the ships, but the lack of firm confirmation leaves it open to interpretation that a contemporary U.S.S. Enterprise exists in the 32nd century.