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 Proxima b could harbor a small dayside surface ocean despite its weak instellation. We present the first 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/or depression of the freezing point by salinity. Elevated greenhouse gas concentrations do not necessarily produce more open ocean because of dynamical regime transitions between a state with an equatorial Rossby-Kelvin wave pattern and a state with a day-night circulation. For an evolutionary path leading to a highly saline ocean, Proxima b could be an inhabited, mostly open ocean planet with halophilic life. A freshwater ocean produces a smaller liquid region than does an Earth salinity ocean. An ocean planet in 3:2 spin-orbit resonance has a permanent tropical waterbelt for moderate eccentricity. A larger versus smaller area of surface liquid water for similar equilibrium temperature may be distinguishable by using the amplitude of the thermal phase curve. Simulations of Proxima Centauri b may be a model for the habitability of weakly irradiated planets orbiting slightly cooler or warmer stars, for example, in the TRAPPIST-1, LHS 1140, GJ 273, and GJ 3293 systems.