A search for the standard model Higgs boson produced in association with a top-quark pair, $t\overline{t}H$, is presented. The analysis uses $36.1\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of $pp$ collision data at $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$ collected with the ATLAS detector at the Large Hadron Collider in 2015 and 2016. The search targets the $H\ensuremath{\rightarrow}b\overline{b}$ decay mode. The selected events contain either one or two electrons or muons from the top-quark decays, and are then categorized according to the number of jets and how likely these are to contain $b$-hadrons. Multivariate techniques are used to discriminate between signal and background events, the latter being dominated by $t\overline{t}+\text{jets}$ production. For a Higgs boson mass of 125 GeV, the ratio of the measured $t\overline{t}H$ signal cross-section to the standard model expectation is found to be $\ensuremath{\mu}=0.8{4}_{\ensuremath{-}0.61}^{+0.64}$. A value of $\ensuremath{\mu}$ greater than 2.0 is excluded at 95% confidence level (C.L.) while the expected upper limit is $\ensuremath{\mu}<1.2$ in the absence of a $t\overline{t}H$ signal.