We study billiard dynamics inside an ellipse for which the axes lengths are periodically in time and an O(\δ)-small quartic polynomial is added to the boundary. In this situation the energy of the in the billiard is no longer conserved. We show a Fermi acceleration such system: there exists a billiard trajectory on which the energy tends to. The construction is based on the analysis of dynamics in the phase near a homoclinic intersection of the stable and unstable manifolds of normally hyperbolic invariant cylinder \Λ, parameterised by the and time, that corresponds to the motion along the major axis of the. The proof depends on the reduction of the billiard map near the channel to an iterated function system comprised by the shifts along Hamiltonian flows defined on \Λ. The two flows approximate the-called inner and scattering maps, which are basic tools that arise in the of the Arnold diffusion; the scattering maps defined by the projection the strong stable and strong unstable foliations Wss,uu of the and unstable invariant manifolds Ws,u(\Λ) at the homoclinic. Melnikov type calculations imply that the behaviour of the scattering in this problem is quite unusual: it is only defined on a small subset of\Λ that shrinks, in the large energy limit, to a set of parallel lines$t=const$ as \δ\→ 0.