Some dynamical and chaotic properties are studied for a classical particle bouncing between two rigid walls, one of which is fixed and the other moves in time, in the presence of an external field. The system is a hybrid, behaving not as a purely Fermi–Ulam model, nor as a bouncer, but as a combination of the two. We consider two different kinds of motion of the moving wall: (i) periodic and (ii) random. The dynamics of the model is studied via a two-dimensional nonlinear area-preserving map. We confirm that, for periodic oscillations, our model recovers the well-known results of the Fermi–Ulam model in the limit of zero external field. For intense external fields, we establish the range of control parameters values within which invariant spanning curves are observed below the chaotic sea in the low energy domain. We characterize this chaotic low energy region in terms of Lyapunov exponents. We also show that the velocity of the particle, and hence also its kinetic energy, grow according to a power law when the wall moves randomly, yielding clear evidence of Fermi acceleration