As an example of a Hamiltonian ratchet in two-dimensional space but without driving, we study a spatially periodic billiard immersed in a uniform stationary magnetic field. Our billiard consists of a straight waveguide with equidistant walls attached perpendicularly to one side. The magnetic field is sufficient to break all symmetries relevant for transport, an external driving therefore is unnecessary. In the low- and high-field limits, the motion becomes respectively pseudo-integrable and integrable; in both cases no transport is possible. In the medium-field regime, surfaces of section reveal mixed phase-space structures. Chaotic components support diffusive transport only in one direction, while regular components can support ballistic transport in both directions. We analyze the dynamical mechanisms underlying directed transport in terms of stable periodic orbits.