  Matter-wave bright solitons are predicted to reflect from a purely attractive
potential well although they are macroscopic objects with classical
particle-like properties. The non-classical reflection occurs at small
velocities and a pronounced switching to almost perfect transmission above a
critical velocity is found, caused by nonlinear mean-field interactions. Full
numerical results from the nonlinear Schr\"{o}dinger equation are complimented
by a two-mode variational calculation to explain the predicted effect, which
can be used for velocity filtering of solitons. The experimental realization
with laser-induced potentials or two-component Bose-Einstein condensates is
suggested.
