We observed discrete fluxon-flow resonance steps in high magnetic fields in a stack of Josephson junctions with lateral size of $1.5{ imes}17{mu}m^2$. The measurement sample was prepared by sandwiching a stack of $Bi_2Sr_2CaCu_2O_{8+x}$ intrinsic Josephson junctions between two Au electrodes by using the double-side-cleaving technique. This technique allowed us to isolate the intrinsic Josephson junction structures from the inductive interference of the basal stack. The resonance steps observed are in good agreement with the collective Josephson fluxon dynamics that are in resonance with the plasma oscillation modes inside the stacked Josephson junctions.