The first measurement of the CP asymmetry of the decay rate (A(CP)) and the CP average (Sigma A(FB)) and CP asymmetry (Delta A(FB)) of the forward-backward asymmetry in the muon system of Lambda(+)(c) -> p mu(+) mu(-) decays is reported. The measurement is performed using a data sample of proton-proton collisions, recorded by the LHCb experiment from 2016 to 2018 at a center-of-mass energy of 13 TeV, which corresponds to an integrated luminosity of 5.4 fb(-1). The asymmetries are measured in two regions of dimuon mass near the.-meson mass peak. The dimuon-mass integrated results are A(CP) = (-1.1 +/- 4.0 +/- 0.5)%, Sigma A(FB) = (3.9 +/- 4.0 +/- 0.6)%, Delta A(FB) = (3.1 +/- 4.0 +/- 0.4)%, where the first uncertainty is statistical and the second systematic. The results are consistent with the conservation of CP symmetry and the Standard Model expectations.