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 proton-proton (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 σ(pp→A^{'}→χ_{1}χ_{2}) and the decay branching fraction B(χ_{2}→χ_{1}μ^{+}μ^{-}), where A^{'} is a dark photon and χ_{1} and χ_{2} are states in the dark sector with near mass degeneracy. This is the first dedicated collider search for inelastic dark matter.