Inclusive and differential cross-sections for the production of a top-quark pair in association with a photon are measured with proton-proton collision data corresponding to an integrated luminosity of 36.1 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$, collected by the ATLAS detector at the LHC in 2015 and 2016 at a centre-of-mass energy of 13 TeV. The measurements are performed in single-lepton and dilepton final states in a fiducial volume. Events with exactly one photon, one or two leptons, a channel-dependent minimum number of jets, and at least one $b$-jet are selected. Neural network algorithms are used to separate the signal from the backgrounds. The fiducial cross-sections are measured to be 521 $\pm$ 9(stat.) $\pm$ 41(sys.) fb and 69 $\pm$ 3(stat.) $\pm$ 4(sys.) fb for the single-lepton and dilepton channels, respectively. The differential cross-sections are measured as a function of photon transverse momentum, photon absolute pseudorapidity, and angular distance between the photon and its closest lepton in both channels, as well as azimuthal opening angle and absolute pseudorapidity difference between the two leptons in the dilepton channel. All measurements are in agreement with the theoretical predictions.