A search is conducted for new resonant and non-resonant high-mass phenomena in dielectron and dimuon final states. The search uses 36.1 fb⁻¹ of proton-proton collision data, collected at √s=13 TeV by the ATLAS experiment at the LHC in 2015 and 2016. No significant deviation from the Standard Model prediction is observed. Upper limits at 95% credibility level are set on the cross-section times branching ratio for resonances decaying into dileptons, which are converted to lower limits on the resonance mass, up to 4.1 TeV for the E₆-motivated Z′_χ. Lower limits on the qqℓℓ contact interaction scale are set between 2.4 TeV and 40 TeV, depending on the model.