A search for a heavy neutral Higgs boson, $A$, decaying into a $Z$ boson and another heavy Higgs boson, $H$, is performed using a data sample corresponding to an integrated luminosity of 36.1 fb$^{-1}$ from proton--proton collisions at $\sqrt{s} = 13$ TeV recorded in 2015 and 2016 by the ATLAS detector at the Large Hadron Collider. The search considers the $Z$ boson decaying to electrons or muons and the $H$ boson into a pair of $b$-quarks. No evidence for the production of an $A$ boson is found. Considering each production process separately, the 95% confidence-level upper limits on the $pp\rightarrow A\rightarrow ZH$ production cross-section times the branching ratio $H\rightarrow bb$ are in the range of 14--830 fb for the gluon--gluon fusion process and 26--570 fb for the $b$-associated process for the mass ranges 130--700 GeV of the $H$ boson and process for the mass ranges 130--700 GeV of the $H$ boson and 230--800 GeV of the $A$ boson. The results are interpreted in the context of the two-Higgs-doublet model.

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