Given a projective Finsler metric in a convex domain in the projective plane, that is, a metric in which geodesics are straight lines, consider the respective Finsler billiard system. Choose a generic point inside the table and consider the billiard trajectories that start at this point and undergo $N$ reflection off the boundary. The envelope of the resulting 1-parameter family of straight lines is the $N$th caustic by reflection. We prove that, for every $N$, it has at least four cusps, generalizing a similar result for Euclidean metric, obtained recently jointly with G. Bor.