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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 $v = 0.01-10~\mathrm{a.u.}$.   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 electronic excitations 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 exponent $\sim 1.5$) in the velocity range $v = 0.1-0.3~\mathrm{a.u.}$ that we associate to the copper crystalline electronic band structure.  The results are rationalized by simple band models in separate regimes.