  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$_\oplus$ with an equilibrium temperature of $\sim
620\,$K. We successfully detect a reflex motion of the star and establish 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 use and compare
different methods to reduce and analyse those data. We report a 5.5-$\sigma$
detection of the planetary signal, giving a mass of $5.5 \pm 1.0\,$M$_\oplus$
and a radius of $2.03 \pm 0.18\,$R$_\oplus$. We derive a mean equilibrium
planet temperature of $590 \pm 90\,$K. The mean density of this small planet of
$3.6 \pm 1.1\,$g$\,$cm$^{-3}$ is similar (1.7-$\sigma$ lower) than that of the
Earth. It leads to a non-negligible fraction of volatiles in its atmosphere
with $f_{H,He}=0.28^{+0.46}_{-0.23}$% or $f_\text{water}=23 \pm 12$%.
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/sub-Neptune-like
planets.
