Static and dynamic magnetic properties of lightly P-substituted BaFe2(As1-xPx)(2) were systematically investigated by P-31 NMR. The averaged internal magnetic field at the P site in the zero-temperature limit evaluated from the broadening of NMR spectra in the antiferromagnetic (AFM) phase is gradually suppressed toward x similar to 0.35 with increasing x, which provides definitive evidence for the existence of an AFM quantum critical point (QCP) at x similar to 0.35. The location of the AFM QCP is consistent with the previous estimation from temperature dependence of spin dynamics in the normal state, and the superconducting transition temperature T-c takes the maximum around the QCP. Our experiments, revealing a signature of a QCP extending up to room temperature, establish BaFe2(As1-xPx)(2) as one of the most accessible systems for unraveling the nature of quantum criticality and the relationship between AFM quantum criticality and unconventional superconductivity.