The production of single-charmed baryons ${\Lambda }_c^{+}{\Lambda }_c^{-}$, ${\Lambda }_c^{+}{\Sigma }_c^{-}+H.c.$, and ${\Sigma }_c^{+}{\Sigma }_c^{-}$ in $p\overline{p}$ collisions is studied in the framework of a constituent quark model that has satisfactorily described the $N\overline{N}$ system and the strangeness production $p\overline{p}\to \Lambda \overline{\Lambda }$, $\Lambda \overline{\Sigma }$, and $\Sigma \overline{\Sigma }$ processes. Predictions on the total cross sections are analyzed for different approaches to the underlying $n\overline{n}\to c\overline{c}$ process, mediated by one-gluon annihilation diagrams. The results indicate that the cross section is on the order of 1 nb between 10 and 14 GeV for ${\Lambda }_c^{+}{\Lambda }_c^{-}$ and ${\Lambda }_c^{+}{\Sigma }_c^{-}$ channels and around 0.01–0.1 nb for ${\Sigma }_c^{+}{\Sigma }_c^{-}$. These estimations can be relevant for their future search in facilities like PANDA.