This paper presents a search for new heavy particles decaying into a pair of top quarks using 139 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of proton--proton collision data recorded at a centre-of-mass energy of $s=13$ TeV with the ATLAS detector at the Large Hadron Collider. The search is performed using events consistent with pair production of high-transverse-momentum top quarks and their subsequent decays into the fully hadronic final states. The analysis is optimized for resonances decaying into a $tt¯$ pair with mass above 1.4 TeV, exploiting a dedicated multivariate technique with jet substructure to identify hadronically decaying top quarks using large-radius jets and evaluating the background expectation from data. No significant deviation from the background prediction is observed. Limits are set on the production cross-section times branching fraction for the new $Z\prime$ boson in a topcolor-assisted-technicolor model. The $Z\prime$ boson masses below 3.9 and 4.7 TeV are excluded at 95% confidence level for the decay widths of 1% and 3%, respectively.