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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%$\mathrm{H^+}$ ion  in a periodic crystal. %$\mathrm{Cu}$ crystal.   These 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 $v = 0.1-0.3~\mathrm{a.u.}$ that 0.1-0.3~\mathrm{a.u.}$, which  we associate to the copper crystalline electronic band structure. The results are rationalized by simple band models in 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 and non-linear effects.