The first measurement of the production rate of ${\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}$ baryons in $pp$ collisions relative to that of ${\mathrm{\ensuremath{\Lambda}}}_{b}^{0}$ baryons is reported, using data samples collected by the LHCb experiment, and corresponding to integrated luminosities of 1, 2 and $1.6\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ at $\sqrt{s}=7$, 8 and 13 TeV, respectively. In the kinematic region $2<\ensuremath{\eta}<6$ and ${p}_{\mathrm{T}}<20\text{ }\text{ }\mathrm{GeV}/c$, we measure $\frac{{f}_{{\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}}}{{f}_{{\mathrm{\ensuremath{\Lambda}}}_{b}^{0}}}\frac{\mathcal{B}({\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}\ensuremath{\rightarrow}J/\ensuremath{\psi}{\mathrm{\ensuremath{\Xi}}}^{\ensuremath{-}})}{\mathcal{B}({\mathrm{\ensuremath{\Lambda}}}_{b}^{0}\ensuremath{\rightarrow}J/\ensuremath{\psi}\mathrm{\ensuremath{\Lambda}})}=(10.8\ifmmode\pm\else\textpm\fi{}0.9\ifmmode\pm\else\textpm\fi{}0.8)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$ [$\sqrt{s}=7,8\text{ }\text{ }\mathrm{TeV}$], $\frac{{f}_{{\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}}}{{f}_{{\mathrm{\ensuremath{\Lambda}}}_{b}^{0}}}\frac{\mathcal{B}({\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}\ensuremath{\rightarrow}J/\ensuremath{\psi}{\mathrm{\ensuremath{\Xi}}}^{\ensuremath{-}})}{\mathcal{B}({\mathrm{\ensuremath{\Lambda}}}_{b}^{0}\ensuremath{\rightarrow}J/\ensuremath{\psi}\mathrm{\ensuremath{\Lambda}})}=(13.1\ifmmode\pm\else\textpm\fi{}1.1\ifmmode\pm\else\textpm\fi{}1.0)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$ [$\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$], where ${f}_{{\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}}$ and ${f}_{{\mathrm{\ensuremath{\Lambda}}}_{b}^{0}}$ are the fragmentation fractions of $b$ quarks into ${\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}$ and ${\mathrm{\ensuremath{\Lambda}}}_{b}^{0}$ baryons, respectively; $\mathcal{B}$ represents branching fractions; and the uncertainties are due to statistical and experimental systematic sources. The values of ${f}_{{\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}}/{f}_{{\mathrm{\ensuremath{\Lambda}}}_{b}^{0}}$ are obtained by invoking SU(3) symmetry in the ${\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}\ensuremath{\rightarrow}J/\ensuremath{\psi}{\mathrm{\ensuremath{\Xi}}}^{\ensuremath{-}}$ and ${\mathrm{\ensuremath{\Lambda}}}_{b}^{0}\ensuremath{\rightarrow}J/\ensuremath{\psi}\mathrm{\ensuremath{\Lambda}}$ decays. Production asymmetries between ${\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}$ and ${\overline{\mathrm{\ensuremath{\Xi}}}}_{b}^{+}$ baryons are also reported. The mass of the ${\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}$ baryon is also measured relative to that of the ${\mathrm{\ensuremath{\Lambda}}}_{b}^{0}$ baryon, from which it is found that $m({\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}})=5796.70\ifmmode\pm\else\textpm\fi{}0.39\ifmmode\pm\else\textpm\fi{}0.15\ifmmode\pm\else\textpm\fi{}0.17\text{ }\text{ }\mathrm{MeV}/{c}^{2}$, where the last uncertainty is due to the precision on the known ${\mathrm{\ensuremath{\Lambda}}}_{b}^{0}$ mass. This result represents the most precise determination of the ${\mathrm{\ensuremath{\Xi}}}_{b}^{\ensuremath{-}}$ mass.