Searches for high-mass resonances in the dijet invariant mass spectrum with one or two jets identified as $b$-jets are performed using an integrated luminosity of $3.2$ fb$^{-1}$ of proton--proton collisions with a centre-of-mass energy of $\sqrt{s}=13$ TeV recorded by the ATLAS detector at the Large Hadron Collider. No evidence of anomalous phenomena is observed in the data, which are used to exclude, at 95% credibility level, excited $b^{*}$ quarks with masses from 1.1 TeV to 2.1 TeV and leptophobic $Z'$ bosons with masses from 1.1 TeV to 1.5 TeV. Contributions of a Gaussian signal shape with effective cross sections ranging from approximately 0.4 to 0.001 pb are also excluded in the mass range 1.5-5.0 TeV.