A search for new phenomena in $t\bar{t}$ final states with additional heavy-flavour jets has been carried out using 13.2 fb$^{−1}$ of data in $pp$ collisions at $\sqrt{s} = 13$ TeV with the ATLAS detector at the LHC. The search targets a variety of signals, including the pair production of vector-like top quarks; four-top-quark production in several new physics scenarios. Data are analysed in the lepton-plus-jets final state as well as the jets-plus-$E^{\mathrm{miss}}_{\mathrm{T}}$ final state. The search exploits the events with the high multiplicity of $b$-jets, the high scalar sum of transverse momenta of all final state objects, and the presence of boosted hadronically-decaying resonances reconstructed as large-radius jets.