The development of time-dependent mean-field theory (TDMF) for the treatment of ion-metal interactions is detailed. By allowing for the time-dependent, nonlinear response of the conduction electrons, TDMF provides a self-consistent description that is free of the adiabatic and linear response approximations that have conventionally been used to treat dynamical processes in simple metals. We present the first results of three-dimensional simulations of a bare proton passing through a thin metallic foil. The nonlinear-induced electron density, dynamical screening potential, and electronic stopping power are all displayed as functions of time for several proton velocities ranging from one-half to eight times the Fermi velocity of a simple metal (sodium). We find that a sizable induced density forms behind the proton and that this density is carried along in the wake of the proton with very little dispersion as it traverses the foil. At proton velocities comparable to or above the Fermi velocity, these wake-riding electrons are shaken off as the proton passes through the rear surface of the foil. We find no evidence that the proton forms a stable hydrogen atom as it traverses the foil. At the velocities studied, the conduction electrons provide a weak, asymmetric screening of the proton, with some regions behind the proton actually being overscreened at the higher velocities. A comparison of our results with a standard linear response treatment of the problem reveals both qualitative and quantitative differences in the calculated time-dependent electron density and screening length. We find that the basic assumption underlying the linear response approximation is not justified in this case, due to the strongly nonlinear nature of the conduction electrons' response. These results are illustrative of the kinds of calculations that can be carried out with the simulation package that we have developed and we describe several applications that are planned for the near future. Several innovations in numerical technique, developed in the course of this work, are also detailed.