A search for the rare leptonic decay $${{{B} ^+}} \!\rightarrow {\mu ^+} {\mu ^-} {\mu ^+} {{\upnu } _\mu } $$ is performed using proton-proton collision data corresponding to an integrated luminosity of $$4.7 \, {\hbox {fb}}^{-1}$$ collected by the LHCb experiment. The search is carried out in the region where the lowest of the two $${\mu ^+\mu ^-} $$ mass combinations is below $$980 {\mathrm {\,MeV\!/}c^2} $$ . The data are consistent with the background-only hypothesis and an upper limit of $$1.6 \times 10^{-8}$$ at 95% confidence level is set on the branching fraction in the stated kinematic region.