A bstract 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 −1 of pp collisions at $$ \sqrt{s}=13 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mi>s</mml:mi> </mml:msqrt> <mml:mo>=</mml:mo> <mml:mn>13</mml:mn> </mml:math> 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 m T &gt; 0.98 (0.99) TeV and m B &gt; 1.00 (1.01) TeV respectively. Limits on the production of the vector-like T 5/3 -quark are also derived considering both pair and single production; in the former case the lower limit on the mass of the T 5/3 -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 → 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.