  GN-108036 is a star-forming galaxy at $z=7.21$, and one of the most distant
known sources in the Northern hemisphere. Based on observations from the
NOrthern Extended Millimeter Array (NOEMA), here we report the tentative
detection of the [CII] line at $\approx4\sigma$ significance. The integrated
[CII] line emission is spatially offset about $\sim4$ kpc from the rest-frame
ultraviolet (UV) emission. The total [CII] luminosity ($L_{\rm
[CII]}=2.7\times10^8~L_{\odot}$) is consistent with the relation between [CII]
luminosity and star formation rate (SFR) observed in nearby and high-$z$ star
forming galaxies. More interestingly, the [CII] line is blueshifted with
respect to the Ly$\alpha$ line by $980\pm10$ km s$^{-1}$. If confirmed, this
corresponds to the largest velocity offset reported to date between the
Ly$\alpha$ line and a non-resonant line at $z\gtrsim6$. According to trends
observed in other high redshift galaxies, the large Ly$\alpha$ velocity offset
in GN-108036 is consistent with its low Ly$\alpha$ equivalent width and high UV
absolute magnitude. Based on Ly$\alpha$ radiative transfer models of expanding
shells, the large Ly$\alpha$ velocity offset in GN-108036 could be interpreted
as the presence of a large column density of hydrogen gas, and/or an outflow
with a velocity of $v_{\rm out}\sim\Delta v_{\rm Ly \alpha}/2\sim500$ km
s$^{-1}$. We also report the 3$\sigma$ detection of a potential galaxy
companion located $\sim30$ kpc east of GN-108036, at a similar systemic
velocity, and with no counterpart rest-frame UV emission.
