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A search for the charmless baryonic decay B+→Λ¯pp¯p is performed using proton-proton collision data recorded by the LHCb experiment, corresponding to an integrated luminosity of 5.4 fb-1. The branching fraction for this decay is measured for the first time relative to that of the topologically similar decay B+→J/ψK+, with J/ψ→Λ¯pK-. The branching fraction is measured to be B(B+→Λ¯pp¯p)=(2.15±0.35±0.12±0.28)×10-7, where the first uncertainty is statistical, the second is systematic, and the third arises from the uncertainty in the normalization channel branching fraction. The CP asymmetry is measured to be ACP=(5.4±15.6±2.4)%, where the uncertainties are statistical and systematic. The background-subtracted invariant-mass distributions of Λ¯p and p¯p pairs exhibit pronounced enhancements at both kinematic thresholds, in contrast to a uniform phase-space distribution.