A search for $C!P$ violation in the decay $Lambda^0_b to p K^- mu^+ mu^-$ is presented. This decay is mediated by flavour-changing neutral-current transitions in the Standard Model and is potentially sensitive to new sources of $C!P$ violation. The study is based on a data sample of proton-proton collisions recorded with the LHCb experiment, corresponding to an integrated luminosity of $3$ $mathrm{fb}^{-1}$. The $Lambda^0_b to p K^- mu^+ mu^-$ decay is observed for the first time, and two observables that are sensitive to different manifestations of $C!P$ violation are measured, $Deltamathcal{A}_{C!P} equiv mathcal{A}_{C!P}(Lambda^0_b to p K^- mu^+ mu^-)-mathcal{A}_{C!P}(Lambda^0_bto pK^- J/psi)$ and $a_{C!P}^{widehat{T}-odd}$, where the latter is based on asymmetries in the angle between the $mu^+mu^-$ and $p K^-$ decay planes. These are measured to be $Deltamathcal{A}_{C!P} = (-3.5 pm 5.0(stat) pm 0.2(syst))times 10^{-2}$, $a_{C!P}^{widehat{T}-odd} = (1.2 pm 5.0(stat) pm 0.7(syst))times 10^{-2}$, and no evidence for $C!P$ violation is found.
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