In order to identify the pairing symmetry with chirality, we study site-selective NMR in chiral p-wave superconductors. We calculate local nuclear relaxation rate T-1(-1) in the vortex lattice state by Eilenberger theory, including the applied magnetic field dependence. We find that T-1(-1) in the NMR resonance line shape is different between two chiral states p +/-(= p(x)+/- ip(y)), depending on whether the chirality is parallel or antiparallel to the vorticity. Anomalous suppression of T-1(-1) occurs around the vortex core in the chiral p(-) wave due to the negative coherence term coming from the odd-frequency s-wave Cooper pair induced around the vortex with Majorana state.