A study of ${\mathrm{\ensuremath{\Lambda}}}_{b}^{0}$ baryon decays to the $Dp{K}^{\ensuremath{-}}$ final state is presented based on a proton-proton collision data sample corresponding to an integrated luminosity of $9\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ collected with the LHCb detector. Two ${\mathrm{\ensuremath{\Lambda}}}_{b}^{0}$ decays are considered, ${\mathrm{\ensuremath{\Lambda}}}_{b}^{0}\ensuremath{\rightarrow}Dp{K}^{\ensuremath{-}}$ with $D\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}$ and $D\ensuremath{\rightarrow}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$, where $D$ represents a superposition of ${D}^{0}$ and ${\overline{D}}^{0}$ states. The latter process is expected to be suppressed relative to the former, and is observed for the first time. The ratio of branching fractions of the two decays is measured, and the $CP$ asymmetry of the suppressed mode, which is sensitive to the Cabibbo-Kobayashi-Maskawa angle $\ensuremath{\gamma}$, is also reported.