Top quarks, the heaviest elementary particles carrying colour charges, are considered to be attractive candidates for probing the quark-gluon plasma produced in relativistic heavy-ion collisions. In proton-lead collisions, top-quark production is expected to be sensitive to nuclear modifications of parton distribution functions at high Bjorken-$x$ values, which are difficult to access experimentally using other available probes. In 2016, the ATLAS experiment recorded proton-lead collisions at a centre-of-mass energy of 8.16 TeV per nucleon pair, corresponding to an integrated luminosity of 165 nb$^{-1}$. In these proceedings, the final measurement of the top-quark pair production in dilepton and lepton+jet decay modes in the proton-lead system with the ATLAS detector is presented. The inclusive cross-section is extracted using a profile-likelihood fit to data distributions in six signal regions. The nuclear modification factor is also measured, and the measurements are found to be in good with theoretical predictions using nuclear parton distribution functions. The relative uncertainty amounts to 9%, making it the most precise top-quark pair cross-section measurement in heavy-ion collisions.