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 (𝑝⁢𝑝) 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⁡(𝑝⁢𝑝→𝐴′→𝜒1⁢𝜒2) and the decay branching fraction ℬ⁡(𝜒2→𝜒1⁢𝜇+⁢𝜇−), where 𝐴′ is a dark photon and 𝜒1 and 𝜒2 are states in the dark sector with near mass degeneracy. This is the first dedicated collider search for inelastic dark matter.