A search for exotic decays of the Higgs boson into a pair of spin-zero particles, H → aa , where the a -boson decay sin to b -quarks promptly or with a mean proper lifetime cτ 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 √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 → WH , ZH and their combination, times the branching ratio of the decay chain H → aa → 4 b . 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 → aa → 4 b 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 0.68 pb, respectively, at cτ a ∼ 0.4 mm.