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The production of inclusive, prompt and non-prompt J/ψ was studied for the first time at midrapidity (−1.37<ycms<0.43) in p−Pb collisions at √sNN =8.16 TeV with the ALICE detector at the LHC. The inclusive J/ψ mesons were reconstructed in the dielectron decay channel in the transverse momentum (pT) interval 0<pT<14 GeV/c and the prompt and non-prompt contributions were separated on a statistical basis for pT>2 GeV/c. The study of the J/ψ mesons in the dielectron channel used for the first time in ALICE online single-electron triggers from the Transition Radiation Detector, providing a data sample corresponding to an integrated luminosity of 689±13μb−1. The proton−proton reference cross section for inclusive J/ψ was obtained based on interpolations of measured data at different centre-of-mass energies and a universal function describing the pT-differential J/ψ production cross sections. The pT-differential nuclear modification factors RpPb of inclusive, prompt, and non-prompt J/ψ are consistent with unity and described by theoretical models implementing only nuclear shadowing.