We present a trajectory planning method for fixed-wing aircraft based on the RRT∗ randomized planning framework. Our main contribution is a trajectory segment design technique that relies on the differential flatness property of the system model. This enables fast analytical evaluation of state and input trajectories. To simplify the integration with an underlying trajectory tracking controller, adjacent trajectory segments are designed to be C² continuous, which translates into at least C⁰ continuity for all relevant system variables. In simulation experiments, real-time performance is achieved for typical application scenarios on hardware equivalent to decent mobile computing units.