Single-photon lidar has emerged as a strong technology for bathymetric measurements. However, its heightened sensitivity additionally makes it susceptible to solar radiation noise, particularly in the green light wavelength where solar radiation is strong, posing challenges for its daytime operation. To address this issue, a single-photon underwater lidar system is proposed and demonstrated. This scheme has these features. 1) Underwater applications not only mitigate the impact of the air-water interface on laser transmission but also significantly attenuate solar radiation reaching the lidar due to the absorption and scattering properties of water. 2) The telescope is designed with a small aperture and narrow field of view to significantly suppress solar radiation. 3) A combination of a narrowband laser and narrowband filter technique is effectively employed to minimize residual solar radiation, thus enabling continuous bathymetric observation capabilities during both day and night. 4) After acquiring the backscattered signal from the bottom, a water depth extraction algorithm utilizing bi-Gaussian fitting is proposed. To demonstrate the robustness of the lidar and the effectiveness of the algorithm, the underwater single-photon lidar system is deployed on a ship to conduct cruise surveys of two bays in the nearshore area, as well as a full-day stationary observation experiment. The lidar measurements are highly consistent with the synchronized sonar observations. The full-day stationary observation experiment showcased its capability to deliver continuous measurements throughout the day and night. These results demonstrate the potential of the system in various applications, including high-precision underwater terrain mapping, obstacle avoidance for underwater platforms, and underwater target imaging.