  An inclusive search for long-lived exotic particles (LLPs) decaying to final
states with a pair of muons is presented. The search uses data corresponding to
an integrated luminosity of 36.6 fb$^{-1}$ collected by the CMS experiment from
the proton-proton collisions at $\sqrt{s}$ = 13.6 TeV in 2022, the first year
of Run 3 of the CERN LHC. The experimental signature is a pair of oppositely
charged muons originating from a common vertex spatially separated from the
proton-proton interaction point by distances ranging from several hundred
$\mu$m to several meters. The sensitivity of the search benefits from new
triggers for displaced dimuons developed for Run 3. The results are interpreted
in the framework of the hidden Abelian Higgs model, in which the Higgs boson
decays to a pair of long-lived dark photons, and of an $R$-parity violating
supersymmetry model, in which long-lived neutralinos decay to a pair of muons
and a neutrino. The limits set on these models are the most stringent to date
in wide regions of lifetimes for LLPs with masses larger than 10 GeV.
