  We present an ab initio study of the electronic stopping power of protons in
copper over a wide range of proton velocities $v = 0.02-10~\mathrm{a.u.}$ 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 $\sim
1.5$) in the velocity range $v = 0.07-0.3~\mathrm{a.u.}$, 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 $v\to 0$ is not as simple as phenomenological models
suggest and it plagued by band-structure effects.
