Measurements of $CP$ asymmetries in charmless three-body decays of $B^\pm$ mesons are reported using proton-proton collision data collected by the LHCb detector, corresponding to an integrated luminosity of 5.9$\,$fb$^{-1}$. The previously observed $CP$ asymmetry in $B^\pm \to \pi^\pm K^+ K^-$ decays is confirmed, and $CP$ asymmetries are observed with a significance of more than five standard deviations in the $B^\pm \to \pi^\pm \pi^+ \pi^-$ and $B^\pm \to K^\pm K^+ K^-$ decays, while the $CP$ asymmetry of $B^\pm \to K^\pm \pi^+ \pi^-$ decays is confirmed to be compatible with zero. The distributions of these asymmetries are also studied as a function of the three-body phase space and suggest contributions from rescattering and resonance interference processes. An indication of the presence of the decays $B^\pm \to \pi^\pm \chi_{c0}(1P)$ in both $B^\pm \to \pi^\pm \pi^+ \pi^-$ and $B^\pm \to \pi^\pm K^+ K^-$ decays is observed, as is $CP$ violation involving these amplitudes.