We predict the occurrence of Fano line shapes in a semiconductor quantum wire with local spin-orbit Rashba coupling. We show that the Rashba interaction acts in a strictly one-dimensional channel as an attractive impurity, leading to the formation of purely bound states. In a quasi-one-dimensional system these bound states couple to the conduction ones through Rashba intersubband mixing, giving rise to pronounced dips in the linear conductance plateaus. We give exact numerical results and propose an approximate model capturing the main ingredients of the effect.