A search for new phenomena in events with two same-charge leptons or three leptons and jets identified as originating from $b$-quarks in a data sample of 36.1 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of $pp$ collisions at $s=13$ TeV recorded by the ATLAS detector at the Large Hadron Collider is reported. No significant excess is found and limits are set on vector-like quark, four-top-quark, and same-sign top-quark pair production. The observed (expected) 95% CL mass limits for a vector-like $T$- and $B$-quark singlet are $mT>0.98$ $(0.99)$ TeV and $mB>1.00$ $(1.01)$ TeV respectively. Limits on the production of the vector-like $T5/3$-quark are also derived considering both pair and single production; in the former case the lower limit on the mass of the $T5/3$-quark is (expected to be) 1.19 (1.21) TeV. The Standard Model four-top-quark production cross-section upper limit is (expected to be) 69 (29) fb. Constraints are also set on exotic four-top-quark production models. Finally, limits are set on same-sign top-quark pair production. The upper limit on $uu\to tt$ production is (expected to be) 89 (59) fb for a mediator mass of 1 TeV, and a dark-matter interpretation is also derived, excluding a mediator of 3 TeV with a dark-sector coupling of 1.0 and a coupling to ordinary matter above 0.31.