CERN-LHC. Differences in the behaviour of matter and antimatter have been observed in $K$ and $B$ meson decays, but not yet in any baryon decay. Such differences are associated with the non-invariance of fundamental interactions under the combined charge-conjugation and parity transformations, known as $CP$ violation. Using data from the LHCb experiment at the Large Hadron Collider, a search is made for $CP$-violating asymmetries in the decay angle distributions of $\Lambda^0_b$ baryons decaying to $p\pi^-\pi^+\pi^-$ and $p\pi^-K^+K^-$ final states. These four-body hadronic decays are a promising place to search for sources of $CP$ violation both within and beyond the Standard Model of particle physics. We find evidence for $CP$ violation in $\Lambda^0_b\to p\pi^-\pi^+\pi^-$ decays with a statistical significance corresponding to 3.3 standard deviations including systematic uncertainties. This represents the first evidence for $CP$ violation in the baryon sector.