Electron charge near atomically sharp corrugations at the surfaces of a solid tends to spill out and smoothen the abrupt variation of the positions of the positively charged atomic nuclei. The reason is that electrons are much less localized than nuclei. This has been discussed already some 70 years ago by Smoluchowski [R. Smoluchowski Phys. Rev. 60 661 (1941)], and the corresponding effect of charge redistribution near surface corrugations bears his name. The Smoluchowski effect focuses on the total electron charge density. It neglects that electrons - in addition to charge - also carry a spin. We discuss spin-dependent electron spill out and demonstrate in a combined theoretical and experimental work that compelling consequences for spin-polarization and spin-dependent transport arise at the edges of magnetic nanostructures due to the spin-dependent Smoluchowski effect. We find a variation of the tunnel magnetoresistance ratio of more than 20% on a length scale of a few atomic diameters.