Unlike the chirality of electrons, the intrinsic chirality of phonons has only surfaced in recent years. Here, we report on the effects of the interaction between electrons and chiral phonons in two-dimensional materials by using a nonperturbative solution. We show that chiral phonons introduce inelastic Umklapp processes resulting in copropagating edge states that coexist with a continuum. Transport simulations further reveal the robustness of the edge states. Our results hint on the possibility of having a metal embedded with hybrid electron-phonon states of matter.