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 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 → 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.[Figure not available: see fulltext.]