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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mn>138</a:mn><a:mtext> </a:mtext><a:mtext> </a:mtext><a:msup><a:mrow><a:mi>fb</a:mi></a:mrow><a:mrow><a:mo>−</a:mo><a:mn>1</a:mn></a:mrow></a:msup></a:mrow></a:math> of proton-proton (<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>p</c:mi><c:mi>p</c:mi></c:math>) 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:mi>σ</e:mi><e:mo stretchy="false">(</e:mo><e:mi>p</e:mi><e:mi>p</e:mi><e:mo stretchy="false">→</e:mo><e:msup><e:mrow><e:mi>A</e:mi></e:mrow><e:mrow><e:mo>′</e:mo></e:mrow></e:msup><e:mo stretchy="false">→</e:mo><e:msub><e:mrow><e:mi>χ</e:mi></e:mrow><e:mrow><e:mn>1</e:mn></e:mrow></e:msub><e:msub><e:mrow><e:mi>χ</e:mi></e:mrow><e:mrow><e:mn>2</e:mn></e:mrow></e:msub><e:mo stretchy="false">)</e:mo></e:mrow></e:math> and the decay branching fraction <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mrow><k:mi mathvariant="script">B</k:mi><k:mo stretchy="false">(</k:mo><k:msub><k:mrow><k:mi>χ</k:mi></k:mrow><k:mrow><k:mn>2</k:mn></k:mrow></k:msub><k:mo stretchy="false">→</k:mo><k:msub><k:mrow><k:mi>χ</k:mi></k:mrow><k:mrow><k:mn>1</k:mn></k:mrow></k:msub><k:msup><k:mrow><k:mi>μ</k:mi></k:mrow><k:mrow><k:mo>+</k:mo></k:mrow></k:msup><k:msup><k:mrow><k:mi>μ</k:mi></k:mrow><k:mrow><k:mo>−</k:mo></k:mrow></k:msup><k:mo stretchy="false">)</k:mo></k:mrow></k:math>, where <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msup><q:mi>A</q:mi><q:mo>′</q:mo></q:msup></q:math> is a dark photon and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>χ</s:mi><s:mn>1</s:mn></s:msub></s:math> and <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi>χ</u:mi><u:mn>2</u:mn></u:msub></u:math> are states in the dark sector with near mass degeneracy. This is the first dedicated collider search for inelastic dark matter. © 2024 CERN, for the CMS Collaboration 2024 CERN