A search is presented for new high-mass resonances decaying into electron or muon pairs. The search uses proton-proton collision data at a centre-of-mass energy of 13 TeV collected by the CMS experiment at the LHC in 2016, corresponding to an integrated luminosity of 36 fb$^{−1}$. Observations are in agreement with standard model expectations. Upper limits on the product of a new resonance production cross section and branching fraction to dileptons are calculated in a model-independent manner. This permits the interpretation of the limits in models predicting a narrow dielectron or dimuon resonance. A scan of different intrinsic width hypotheses is performed. Limits are set on the masses of various hypothetical particles. For the $ {Z}{\mathrm{SSM}}^{\prime}\left({Z}{{}^{\psi}}^{\prime}\right) $ particle, which arises in the sequential standard model (superstring-inspired model), a lower mass limit of 4.50 (3.90) TeV is set at 95% confidence level. The lightest Kaluza-Klein graviton arising in the Randall-Sundrum model of extra dimensions, with coupling parameters k/M$_{Pl}$ of 0.01, 0.05, and 0.10, is excluded at 95% confidence level below 2.10, 3.65, and 4.25 TeV, respectively. In a simplified model of dark matter production via a vector or axial vector mediator, limits at 95% confidence level are obtained on the masses of the dark matter particle and its mediator.