TOI-1695 is a V-mag = 13 M-dwarf star from the northern hemisphere at 45 pc from the Sun, around which a 3.134-day periodic transit signal from a super-Earth candidate was identified in TESS photometry. With a transit depth of 1.3 mmag, the radius of candidate TOI-1695.01 was estimated by the TESS pipeline to be 1.82 R-circle plus with an equilibrium temperature of similar to 620 K. We successfully detected a reflex motion of the star and establish that it is due to a planetary companion at an orbital period consistent with the photometric transit period, thanks to a year-long radial-velocity monitoring of TOI-1695 by the SPIRou infrared spectropolarimeter. We used and compared different methods to reduce and analyze those data. We report a 5.5 sigma detection of the planetary signal, giving a mass of 5.5 +/- 1.0 M-circle plus and a radius of 2.03 +/- 0.18 R-circle plus. We derive a mean equilibrium planet temperature of 590 +/- 90 K. The mean density of this small planet of 3.6 +/- 1.1 g cm(-3) is similar (1.7 sigma lower) than that of the Earth. It leads to a nonnegligible fraction of volatiles in its atmosphere with f(H,He) = 0.28(-0.23)(+0.46)% or f(water) = 23 +/- 12%. TOI-1695 b is a new sub-Neptune planet at the border of the M-dwarf radius valley that can help test formation scenarios for super-Earth and sub-Neptune-like planets.