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 (Jmag = 9.1). Using precision RVs from the Subaru Telescope IRD (InfraRed Doppler) instrument, we derive a semi-amplitude of $3.92^{+0.60}_{-0.58}\:\mbox{m}\:{\mbox{s}^{-1}}$, corresponding to a planet with a minimum mass $m \sin i = 4.00^{+0.53}_{-0.55}\, M_{\oplus }$. 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 $0.05366^{+0.00056}_{-0.00049}\:$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.