Summary: We construct a new class of \((p,q)\)-extended Poincaré supergravity theories in \(2+1\) dimensions as Chern-Simons theories of supersymmetry algebras with both central and automorphism charges. The new theories have the advantage that they are limits of corresponding \((p,q)\) AdS supergravity theories and, for not too large a value of \(N=p+q\), that they have a natural formulation in terms of off-shell superfields, in which context the distinction between theories having the same value of \(N\) but different \((p,q)\) arises because of inequivalent conformal compensator superfields. We also show that, unlike previously constructed N-extended Poincaré supergravity theories, the new \((2,0)\) theory admits conical spacetimes with Killing spinors. Many of our results on \((2,0)\) Poincaré supergravity continue to apply in the presence of coupling to \(N=2\) supersymmetric sigma-model matter.