  Searches for the rare hadronic decays $B^0 \rightarrow p\bar{p}p\bar{p}$ and
$B^0_{s} \rightarrow p\bar{p}p\bar{p}$ are performed using proton-proton
collision data recorded by the LHCb experiment and corresponding to an
integrated luminosity of 9$\rm{fb}^{-1}$. Significances of $9.3\,\sigma$ and
$4.0\,\sigma$, including statistical and systematic uncertainties, are obtained
for the $B^0 \rightarrow p\bar{p}p\bar{p}$ and $B^0_{s} \rightarrow
p\bar{p}p\bar{p}$ signals, respectively. The branching fractions are measured
relative to the topologically similar normalisation decays $B^0 \rightarrow
J/\psi(\rightarrow p\bar{p}) K^{*0}(\rightarrow K^+ \pi^-)$ and $B^0_{s}
\rightarrow J/\psi(\rightarrow p\bar{p}) \phi(\rightarrow K^+ K^-)$. The
branching fractions are measured to be $ \mathcal{B}(B^0 \rightarrow
p\bar{p}p\bar{p}) = ( 2.2 \pm 0.4 \pm 0.1 \pm 0.1 ) \times 10^{-8} $ and $
\mathcal{B}(B^0_{s} \rightarrow p\bar{p}p\bar{p}) = ( 2.3 \pm 1.0 \pm 0.2 \pm
0.1 ) \times 10^{-8}$. In these measurements, the first uncertainty is
statistical, the second is systematic and the third one is due to the external
branching fraction of the normalisation channel.
