In order to 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 with and without non-magnetic impurity scattering in the Born limit and unitary limit. The local T-1(-1) in the NMR resonance line shape is different between two chiral states p(+/-), depending on whether the chirality is parallel or anti-parallel to the vorticity. In the p(-)-wave, anomalous suppression of local T-1(-1) occurs around the vortex core due to the negative coherence term coming from odd-frequency s-wave Cooper pair induced around the vortex. We especially examine the site dependence of the anomalous suppression of local T-1(-1), including the applied magnetic field dependence and the impurity effects. (C) 2016 Elsevier B.V. All rights reserved.