  A search is presented for the pair production of heavy vector-like $B$
quarks, primarily targeting $B$ quark decays into a $W$ boson and a top quark.
The search is based on $36.1$ $fb^{-1}$ of $pp$ collisions at $\sqrt{s}$ = 13
TeV recorded in 2015 and 2016 with the ATLAS detector at the CERN Large Hadron
Collider. Data are analysed in the lepton-plus-jets final state, characterised
by a high-transverse-momentum isolated electron or muon, large missing
transverse momentum, and multiple jets, of which at least one is $b$-tagged. No
significant deviation from the Standard Model expectation is observed. The 95%
confidence level lower limit on the $B$ mass is 1350 GeV assuming a 100%
branching ratio to $Wt$. In the SU(2) singlet scenario, the lower mass limit is
1170 GeV. This search is also sensitive to a heavy vector-like $B$ quark
decaying into other final states ($Zb$ and $Hb$) and thus mass limits on $B$
production are set as a function of the decay branching ratios. The 100%
branching ratio limits are found to be also applicable to heavy vector-like $X$
production, with charge $+$5/3, that decay into $Wt$.
