We report the near-infrared radial velocity (RV) discovery of a super-Earth planet on a 10.77 d orbit around the M4.5 dwarf Ross 508 (J_mag = 9.1). Using precision RVs from the Subaru Telescope IRD (InfraRed Doppler) instrument, we derive a semi-amplitude of , corresponding to a planet with a minimum mass . We find no evidence of significant signals at the detected period in spectroscopic stellar activity indicators or MEarth photometry. The planet, Ross 508 b, has a semi-major axis of au. This gives an orbit-averaged insolation of ≈1.4 times the Earth’s value, placing Ross 508 b near the inner edge of its star’s habitable zone. We have explored the possibility that the planet has a high eccentricity and its host is accompanied by an additional unconfirmed companion on a wide orbit. Our discovery demonstrates that the near-infrared RV search can play a crucial role in finding a low-mass planet around cool M dwarfs like Ross 508.