  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.
