Certain (4, 4)-supersymmetric (1 + 1)-dimensional sigma-models admit a scalar potential. We exhibit models of this type for which there are static particle-like solutions stabilized by a three-vector topological charge Q. These solutions form a special class of (generally time-dependent) "Q-kink" solutions stabilized by a quaternionic charge Q = (Q0, Q), where Q0 is a Noether charge. Q-kinks saturate a Bogomol'nyi-type bound and break half the supersymmetry. As solutions of a four-dimensional field theory they constitute examples of type II supermembranes.