A search for dark matter in events with a displaced nonresonant muon pair and missing transverse momentum is presented. The analysis is performed using an integrated luminosity of 138 fb-1 of protonproton (pp) collision data at a center -of -mass energy of 13 TeV produced by the LHC in 2016-2018. No significant excess over the predicted backgrounds is observed. Upper limits are set on the product of the inelastic dark matter production cross section sigma(pp -> A0 -> chi 1 chi 2) and the decay branching fraction B(chi 2 -> chi 1 mu thorn mu-), where A0 is a dark photon and chi 1 and chi 2 are states in the dark sector with near mass degeneracy. This is the first dedicated collider search for inelastic dark matter.