Three-body \(B^+\to p \overline p K^+\) and \(B^+\to p \overline p \pi^+\) decays are studied using a data sample corresponding to an integrated luminosity of 3.0 \(fb^{-1}\) collected by the LHCb experiment in proton-proton collisions at center-of-mass energies of \(7\) and \(8\) TeV. Evidence of CP violation in the \(B^+\to p \overline p K^+\) decay is found in regions of the phase space, representing the first measurement of this kind for a final state containing baryons. Measurements of the forward-backward asymmetry of the light meson in the \(p\overline p\) rest frame yield \(A_{\mathrm{FB}}(p \overline p K^+,~m_{p\overline p}<2.85\mathrm{\,Ge\kern -0.1em V\!/}c^2)=0.495\pm0.012~(\mathrm{stat})\pm0.007~(\mathrm{syst})\) and \(A_{\mathrm{FB}}(p\overline p \pi^+,~m_{p\overline p}<2.85\mathrm{\,Ge\kern -0.1em V\!/}c^2)=-0.409\pm0.033~(\mathrm{stat})\pm0.006~(\mathrm{syst})\). In addition, the branching fraction of the decay \(B^+\to\kern 0.1em\overline{\kern -0.1em\Lambda}(1520)p\) is measured to be \(\mathcal{B}(B^+ \to \kern 0.1em\overline{\kern -0.1em\Lambda}(1520)p)=(3.15\pm0.48~(\mathrm{stat})\pm0.07~(\mathrm{syst})\pm0.26~(\mathrm{BF}))\times 10^{-7}\), where BF denotes the uncertainty on secondary branching fractions.