  We consider a dark sector containing a pair of almost degenerate states
coupled to the Standard Model through a dark photon mediator. This set-up
constitutes a simple realization of the inelastic dark matter scenario. The
heaviest dark state is long-lived, in the limit of a small kinetic mixing among
the dark photon and the Standard Model hypercharge gauge boson, and/or of a
small mass splitting among the dark states. We study the prospects for
detection of this scenario at proposed LHC experiments dedicated to search for
long-lived particles, namely FASER, MATHUSLA, CODEX-b, AL3X, MAPP, ANUBIS and
FACET. We consider both the cases of fermionic and scalar inelastic dark
matter. We show that these experimental facilities can probe unexplored regions
of the parameter space of this model, and we highlight their complementary
roles.
