Topological defects are found ubiquitously in various kinds of matter, such as vortices in type-II superconductors, and magnetic skyrmions in chiral ferromagnets. While knowledge on the static behavior of magnetic skyrmions is accumulating steadily, their dynamics under forced flow is still a widely open issue. Here, we report the deformation of the moving magnetic skyrmion lattice in MnSi under electric current flow observed using small-angle neutron scattering. A spatially inhomogeneous rotation of the skyrmion lattice, with an inverse rotation sense for opposite sample edges, is observed for current densities greater than a threshold value j_t ~ 1 MA m⁻² (10⁶ A m⁻²). Our result show that skyrmion lattices under current flow experience significant friction near the sample edges due to pinning, this being a critical effect that must be considered for anticipated skyrmion-based applications at the nanoscale.