A search for the non-resonant decays B 0 → μ + μ − μ + μ − and B 0 → μ + μ − μ + μ − 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 fb−1 collected at centre-of-mass energies of 7 and 8 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 $$ \mathrm{\mathcal{B}}\left({B}_s^0\to {\mu}^{+}{\mu}^{-}{\mu}^{+}{\mu}^{-}\right)<2.5\times 1{0}^{-9}, $$ $$ \mathrm{\mathcal{B}}\left({B}^0\to {\mu}^{+}{\mu}^{-}{\mu}^{+}{\mu}^{-}\right)<6.9\times 1{0}^{-10}. $$