Solitons are commonly known as waves that propagate without dispersion. Here, we show that they can occur for driven overdamped Brownian dynamics of hard spheres in periodic potentials at high densities. The solitons manifest themselves as periodic sequences of different assemblies of particles moving in the limit of zero noise, where transport of single particles is not possible. They give rise to particle currents at even low temperature that appear in bandlike structures around certain hard-sphere diameters. At high temperatures, the bandlike structures are washed out by the noise, but the particle transport is still dominated by the solitons. All these predicted features should occur in a broad class of periodic systems and are amenable to experimental tests.