Using small-angle neutron scattering (SANS), we investigate the deformation of the magnetic skyrmion lattice in bulk single-crystalline MnSi under electric current flow. A significant broadening of the skyrmion-lattice-reflection peaks was observed in the SANS pattern for current densities greater than a threshold value j_t ~ 1 MA/m^2 (10^6 A/m^2). We show this peak broadening to originate from a spatially inhomogeneous rotation of the skyrmion lattice, with an inverse rotation sense observed for opposite sample edges aligned with the direction of current flow. The peak broadening (and the corresponding skyrmion lattice rotations) remain finite even after switching off the electric current. These results indicate that skyrmion lattices under current flow experience significant friction near the sample edges, and plastic deformation due to pinning effects, these being important factors that must be considered for the anticipated skyrmion-based applications in chiral magnets at the nanoscale.