  We study the entanglement dynamics generated by quantum quenches in the
quantum cellular automaton Rule $54$. We consider the evolution from a recently
introduced class of solvable initial states. States in this class relax
(locally) to a one-parameter family of Gibbs states and the thermalisation
dynamics of local observables can be characterised exactly by means of an
evolution in space. Here we show that the latter approach also gives access to
the entanglement dynamics and derive exact formulas describing the asymptotic
linear growth of all R\'enyi entropies in the thermodynamic limit and their
eventual saturation for finite subsystems. While in the case of von Neumann
entropy we recover exactly the predictions of the quasiparticle picture, we
find no physically meaningful quasiparticle description for other R\'enyi
entropies. Our results apply to both homogeneous and inhomogeneous quenches.
