Neutral pion (π⁰) and η meson production cross sections were measured up to unprecedentedly high transverse momenta (p_T) in p–Pb collisions at s_NN=8.16TeV. 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<p_T<200 GeV/c and 1.0<p_T<50 GeV/c, respectively. The respective nuclear modification factor (R_pPb) is presented for p_T up to of 200 and 30 GeV/c, where the former was achieved by extending the π⁰ measurement in pp collisions at s=8TeV using the merged cluster technique. The values of R_pPb are below unity for p_T<10 GeV/c, while they are consistent with unity for p_T>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 s_NN=5.02TeV.