We present a study of the inclusive production of the $X(4140)$ state with the decay to the $J/\ensuremath{\psi}\ensuremath{\phi}$ final state in hadronic collisions. Based on $10.4\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of $p\overline{p}$ collision data collected by the D0 experiment at the Fermilab Tevatron collider, we report the first evidence for the prompt production of an $X(4140)$ state and find the fraction of $X(4140)$ events originating from $b$ hadrons to be ${f}_{b}=0.39\ifmmode\pm\else\textpm\fi{}0.07(\text{stat})\ifmmode\pm\else\textpm\fi{}0.10(\text{syst})$. The ratio of the nonprompt $X(4140)$ production rate to the ${B}_{s}^{0}$ yield in the same channel is $R=0.19\ifmmode\pm\else\textpm\fi{}0.05(\text{stat})\ifmmode\pm\else\textpm\fi{}0.07(\text{syst})$. The values of the mass $M=4152.5\ifmmode\pm\else\textpm\fi{}1.7(\text{stat}{)}_{\ensuremath{-}5.4}^{+6.2}(\text{syst})\text{ }\mathrm{MeV}$ and width $\mathrm{\ensuremath{\Gamma}}=16.3\ifmmode\pm\else\textpm\fi{}5.6(\text{stat})\ifmmode\pm\else\textpm\fi{}11.4(\text{syst})\text{ }\mathrm{MeV}$ are consistent with previous measurements.