Multiphoton excitation and ionization studies of the Li $23s$ state by 20 ps far-infrared laser pulses originating from the free-electron laser FELIX are presented. The two-photon ionization cross section of highly excited Rydberg $s$ states is small, due to a newly discovered Cooper minimum in the bound-bound $ns\ensuremath{-}{n}^{\ensuremath{'}}p$ transitions. Instead of this lowest-order process, ionization proceeds via a remarkable multiphoton bypass starting with stimulated emission to a lower-lying state. The measurements are interpreted with calculations of the Cooper minimum and a Floquet analysis of the states involved in the bypass.