  We present a comprehensive spectro-photometric analysis of the galaxy GS9422
from the JADES GTO survey located at $z=5.943$, anomalously showing a
simultaneous strong Ly$\alpha$ emission feature and damped Ly$\alpha$
absorption (DLA), based on JWST NIRSpec and NIRCam observations. The best-fit
modelling of the spectral energy distribution (SED) reveals a young, low-mass
(${\rm log}(M_\star/M_{\odot}) = 7.8 \pm 0.01$) galaxy, with a mass-weighted
mean age of the stellar population of $(10.9^{+0.07}_{-0.12})\,$Myr. The
identified strong nebular emission lines suggest a highly ionized ($O_{32} =
59$), low-metallicity ($12+\log({\rm O/H}) = 7.78\pm 0.10$) star-forming galaxy
with a star-formation rate SFR = ($8.2 \pm 2.8$) $\rm M_{\odot}\;yr^{-1}$ over
a compact surface area $A_e = 1.85$ kpc$^{2}$, typical for galaxies at this
epoch. We carefully model the rest-frame UV NIRSpec Prism spectrum around the
Ly$\alpha$ edge, finding that the Ly$\alpha$ emission-line redshift is
consistent with the longer-wavelength recombination lines and an escape
fraction of $f_{\rm esc,Ly\alpha} = 30\%$ but that the broad DLA feature is not
able to converge on the same redshift. Instead, our modelling suggests $z_{\rm
abs}= 5.40 \pm 0.10$, the exact redshift of a newly identified proto-cluster in
nearby projection to the target galaxy. We argue that most of the HI gas
producing the strong Ly$\alpha$ damping wing indeed has to be unassociated with
the galaxy itself, and thus may indicate that we are probing the cold, dense
circumcluster medium of this massive galaxy overdensity. These results provide
an alternative solution to the recent claims of continuum nebular emission or
an obscured active galactic nucleus dominating the rest-frame UV parts of the
spectrum and provide further indications that strong DLAs might preferentially
be associated with galaxy overdensities. [Abridged]
