A search for the non-resonant decays $B^0_s rightarrow mu^{+}mu^{-}mu^{+}mu^{-}$ and $B^0 rightarrow mu^{+}mu^{-}mu^{+}mu^{-}$ is presented. The measurement is performed using the full Run 1 data set collected in proton-proton collisions by the LHCb experiment at the LHC. The data correspond to integrated luminosities of $1$ and $2~mathrm{fb}^{-1}$ collected at centre-of-mass energies of $7$ and $8~mathrm{TeV}$, respectively. No signal is observed and upper limits on the branching fractions of the non-resonant decays at $95%$ confidence level are determined to be mathcal{B}(B^0_s rightarrow mu^{+}mu^{-}mu^{+}mu^{-}) & < 2.5 times 10^{-9} mathcal{B}(B^0 rightarrow mu^{+}mu^{-}mu^{+}mu^{-}) & < 6.9 times 10^{-10}.
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