  Employing X-ray photon correlation spectroscopy we measure the kinetics and
dynamics of a pressure-induced liquid-liquid phase separation (LLPS) in a
water-lysozyme solution. Scattering invariants and kinetic information provide
evidence that the system reaches the phase boundary upon pressure-induced LLPS
with no sign of arrest. The coarsening slows down with increasing quench
depths. The $g_2$-functions display a two-step decay with a gradually
increasing non-ergodicity parameter typical for gelation. We observe fast
superdiffusive ($\gamma \geq 3/2$) and slow subdiffusive ($\gamma < 0.6$)
motion associated with fast viscoelastic fluctuations of the network and a slow
viscous coarsening process, respectively. The dynamics age linear with time
$\tau \propto t_\mathrm{w}$ and we observe the onset of viscoelastic relaxation
for deeper quenches. Our results suggest that the protein solution gels upon
reaching the phase boundary.
