We present an ab initio study of the electronic stopping power of protons in copper over a wide range of proton velocities v = 0.02 − 10 a.u. where we take into account non-linear effects. Time-dependent density functional theory coupled with molecular dynamics is used to study electronic excitations produced by energetic protons. A plane-wave pseudopotential scheme is employed to solve the time-dependent Kohn-Sham equations for a moving ion in a periodic crystal. The electronic excitations and the band structure determine the stopping power of the material and alter the interatomic forces for both channeling and off-channeling trajectories. Our off-channeling results are in quantitative agreement with experiments, and at low velocity they unveil a crossover region of superlinear velocity dependence (with a power of ∼1.5) in the velocity range v = 0.07 − 0.3 a.u., which we associate to the copper crystalline electronic band structure. The results are rationalized by simple band models connecting two separate regimes. We find that the limit of electronic stopping v → 0 is not as simple as phenomenological models suggest and it plagued by band-structure effects.