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Neutral pion (π0) and η meson production cross sections were measured up to unprecedentedly high transverse momenta (pT) in p-Pb collisions at sNN−−−√ = 8.16 TeV. The mesons were reconstructed via their two-photon decay channel in the rapidity interval −1.3<y<0.3 in the ranges of 0.4<pT<200 GeV/c and 1.0<pT<50 GeV/c, respectively. The respective nuclear modification factor (RpPb) is presented for pT up to of 200 and 30 GeV/c, where the former was achieved by extending the π0 measurement in pp collisions at s√ = 8 TeV using the merged cluster technique. The values of RpPb are below unity for pT<10 GeV/c, while they are consistent with unity for pT>10 GeV/c, leaving essentially no room for final state energy loss. The new data provide strong constraints for nuclear parton distribution and fragmentation functions over a broad kinematic range and are compared to model predictions as well as previous results at sNN−−−√ = 5.02 TeV.