We computationally study the transport of active, self-propelled particles suspended in a two-dimensional chaotic flow. The pointlike, spherical particles have their own intrinsic swimming velocity, which modifies the dynamical system so that the particles can break the transport barriers present in the carrier flow. Surprisingly, we find that swimming does not necessarily lead to enhanced particle transport. Small but finite swimming speed can result in reduced transport, as swimmers get stuck for long times in traps that form near elliptic islands in the background flow. Our results have implications for models of transport and encounter rates for small marine organisms.