A search for exotic decays of the Higgs boson into a pair of spin-zero particles, $H \rightarrow aa$, where the $a$-boson decays into $b$-quarks promptly or with a mean proper lifetime $c\tau_a$ up to 6 mm and has a mass in the range of $20-60$ GeV, is presented. The search is performed in events where the Higgs boson is produced in association with a $W$ or $Z$ boson, giving rise to a signature of one or two charged leptons (electrons or muons) and multiple jets from $b$-quark decays. The analysis is based on the dataset of proton-proton collisions at $\sqrt{s} = 13$ TeV recorded in 2015 and 2016 by the ATLAS detector at the CERN Large Hadron Collider, corresponding to an integrated luminosity of $36.1$ fb$^{-1}$. No significant excess of events above the Standard Model background prediction is observed, and 95% confidence-level upper limits are derived for the production cross-sections for $pp \rightarrow WH$, $ZH$ and their combination, times the branching ratio of the decay chain $H \rightarrow aa \rightarrow 4b$. For $a$-bosons which decay promptly, the upper limit on the combination of cross-sections for $WH$ and $ZH$ times the branching ratio of $H \rightarrow aa \rightarrow 4b$ ranges from 3.0 pb for $m_a= 20$ GeV to 1.3 pb for $m_a = 60$ GeV, assuming that the ratio of $WH$ to $ZH$ cross-sections follows the Standard Model prediction. For $a$-bosons with longer proper lifetimes, the most stringent limits are 1.8 pb and 0.6 pb, respectively, at $c\tau_a \sim 0.4$ mm.

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