  In this paper, we construct the first asymmetric strongly interacting massive
particles (SIMP) dark matter (DM) model, where a new vector-like fermion and a
new complex scalar both having nonzero chemical potentials can be asymmetric DM
particles. After the spontaneous breaking of a U(1)$^{}_\textsf{D}$ dark gauge
symmetry, these two particles can have accidental $\mathbb{Z}^{}_4$ charges
making them stable. By adding one more complex scalar as a mediator between the
SIMP DM, the relic density of DM is determined by $3 \to 2$ and two-loop
induced $2 \to 2$ annihilations in this model. On the other hand, the SIMP DM
can maintain kinetic equilibrium with the thermal bath until the DM freeze-out
temperature via the new gauge interaction. Interestingly, this model can have a
bouncing effect on DM, whereby the DM number density rises after the chemical
freeze-out of DM. With this effect, the prediction of the DM self-interacting
cross section in this model can be consistent with astrophysical observations,
and the ratio of the DM energy density to the baryonic matter energy density
can be explained by primordial asymmetries. We also predict the DM-electron
elastic scattering cross section that can be used to test this model in future
projected experiments.
