  The nearby exoplanet Proxima Centauri b will be a prime future target for
characterization, despite questions about its retention of water. Climate
models with static oceans suggest that an Earth-like Proxima b could harbor a
small dayside region of surface liquid water at fairly warm temperatures
despite its weak instellation. We present the first 3-dimensional climate
simulations of Proxima b with a dynamic ocean. We find that an ocean-covered
Proxima b could have a much broader area of surface liquid water but at much
colder temperatures than previously suggested, due to ocean heat transport and
depression of the freezing point by salinity. Elevated greenhouse gas
concentrations do not necessarily produce more open ocean area because of
possible dynamic regime transitions. For an evolutionary path leading to a
highly saline present ocean, Proxima b could conceivably be an inhabited,
mostly open ocean planet dominated by halophilic life. For an ocean planet in
3:2 spin-orbit resonance, a permanent tropical waterbelt exists for moderate
eccentricity. Simulations of Proxima Centauri b may also be a model for the
habitability of planets receiving similar instellation from slightly cooler or
warmer stars, e.g., in the TRAPPIST-1, LHS 1140, GJ 273, and GJ 3293 systems.
