A search for the rare hadronic decay ${B}_{s}^{0}\ensuremath{\rightarrow}p\overline{p}$ is performed using proton-proton collision data recorded by the LHCb experiment at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of $6\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. No evidence of the decay is found and an upper limit on its branching fraction is set at $\mathcal{B}({B}_{s}^{0}\ensuremath{\rightarrow}p\overline{p})<4.4(5.1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}$ at 90% (95%) confidence level; this is currently the world's best upper limit. The decay mode ${B}^{0}\ensuremath{\rightarrow}p\overline{p}$ is measured with very large significance, confirming the first observation by the LHCb experiment in 2017. The branching fraction is determined to be $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}p\overline{p})=(1.27\ifmmode\pm\else\textpm\fi{}0.15\ifmmode\pm\else\textpm\fi{}0.05\ifmmode\pm\else\textpm\fi{}0.04)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}$, where the first uncertainty is statistical, the second is systematic and the third is due to the external branching fraction of the normalization channel ${B}^{0}\ensuremath{\rightarrow}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$. The combination of the two LHCb measurements of the ${B}^{0}\ensuremath{\rightarrow}p\overline{p}$ branching fraction yields $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}p\overline{p})=(1.27\ifmmode\pm\else\textpm\fi{}0.13\ifmmode\pm\else\textpm\fi{}0.05\ifmmode\pm\else\textpm\fi{}0.03)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}$.