<p>It has recently been theoretically predicted and experimentally observed that a soliton resulting from nonlinearity can be pumped across an integer or fractional number of unit cells as a system parameter is slowly varied over a pump period. Nonlinear soliton pumping is now understood as the flow of instantaneous Wannier functions, ruling out the possibility of pumping a soliton across a nonzero number of unit cells over one cycle when a corresponding Wannier function does not exhibit any flow, i.e., when the corresponding Bloch band that the soliton bifurcates from is topologically trivial. Here, we surprisingly find an anomalous nonlinear soliton pump where the displacement of a soliton over one cycle differs from the Chern number of the Bloch band from which the soliton comes. We show that this anomalous behavior arises from a transition of a soliton between different Wannier functions by passing through an intersite-soliton (or dipole-soliton) state. Furthermore, we find a nonlinearity-induced integer-quantized pump of a soliton, allowing a soliton to travel across one unit cell during a pump period, even when the corresponding band is topologically trivial. Our results open the door to studying nonlinearity-induced pumping of solitons.</p>