A search for new heavy particles that decay into top-quark pairs is performed using data collected from proton–proton collisions at a centre-of-mass energy of 13 TeV by the ATLAS detector at the Large Hadron Collider. The integrated luminosity of the data sample is 36.1 fb−1. Events consistent with top-quark pair production are selected by requiring a single isolated charged lepton, missing transverse momentum and jet activity compatible with a hadronic top-quark decay. Jets identified as likely to contain b-hadrons are required to reduce the background from other Standard Model processes. The invariant mass spectrum of the candidate top-quark pairs is examined for local excesses above the background expectation. No significant deviations from the Standard Model predictions are found. Exclusion limits are set on the production cross-section times branching ratio for hypothetical Z′ bosons, Kaluza–Kein gluons and Kaluza–Klein gravitons that decay into top-quark pairs.