  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,
$\Delta\mathcal{A}_{C\!P} \equiv \mathcal{A}_{C\!P}(\Lambda^0_b \to p K^- \mu^+
\mu^-)-\mathcal{A}_{C\!P}(\Lambda^0_b\to pK^- J/\psi)$ 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 $\Delta\mathcal{A}_{C\!P} = (-3.5 \pm 5.0(stat) \pm 0.2(syst))\times
10^{-2}$, $a_{C\!P}^{\widehat{T}-odd} = (1.2 \pm 5.0(stat) \pm 0.7(syst))\times
10^{-2}$, and no evidence for $C\!P$ violation is found.
