The Ly$\alpha$ emission line is one of the most promising probes of cosmic reionisation but isolating the signature of a change in the ionisation state of the IGM is challenging because of intrinsic evolution and internal radiation transfer effects. We present the first study of the evolution of Ly$\alpha$ emitters (LAE) during the epoch of reionisation based on a full radiation-hydrodynamics cosmological simulation that is able to capture both the large-scale process of reionisation and the small-scale properties of galaxies. We predict the Ly$\alpha$ emission of galaxies in the $103$ cMpc$3$ SPHINX simulation at $6\le z\le 9$ by computing the full Ly$\alpha$ radiation transfer from ISM to IGM scales. SPHINX is able to reproduce many observational constraints such as the UV/Ly$\alpha$ luminosity functions and stellar mass functions at z $\ge$ 6 for the dynamical range probed by our simulation ($M1500\gtrsim -18$, $LLy\alpha \lesssim 1042$ erg/s, $M\star \lesssim 109$ M$\mathrm{<mrow\; data-mjx-texclass="ORD">\odot </mrow>}$). As intrinsic Ly$\alpha$ emission and internal Ly$\alpha$ escape fractions barely evolve from $z=6$ to 9, the observed suppression of Ly$\alpha$ luminosities with increasing redshift is fully attributed to IGM absorption. For most observable galaxies ($M1500\lesssim -16$), the Ly$\alpha$ line profiles are slightly shifted to the red due to internal radiative transfer effects which mitigates the effect of IGM absorption. Overall, the enhanced Ly$\alpha$ suppression during reionisation traces the IGM neutral fraction $xHI$ well but the predicted amplitude of this reduction is a strong function of the Ly$\alpha$ peak shift, which is set at ISM/CGM scales. We find that a large number of LAEs could be detectable in very deep surveys during reionisation when $xHI$ is still $\approx 50\%$.