We present a search for Kaluza-Klein (KK) particles predicted by models with universal extra dimensions (UED) using a data set corresponding to an integrated luminosity of $7.3\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$, collected by the D0 detector at a $p\overline{p}$ center-of-mass energy of 1.96 TeV. The decay chain of KK particles can lead to a final state with two muons of the same charge. This signature is used to set a lower limit on the compactification scale of ${R}^{\ensuremath{-}1}>260\text{ }\text{ }\mathrm{GeV}$ in a minimal UED model.