  We analyze the prospects of observing lepton flavour violation in future e-e-
and e+e- linear colliders in scenarios where the gravitino is the lightest
supersymmetric particle, and the stau is the next-to-lightest supersymmetric
particle. The signals consist of multilepton final states with two heavily
ionizing charged tracks produced by the long-lived staus. The Standard Model
backgrounds are very small and the supersymmetric backgrounds can be kept well
under control by the use of suitable kinematical cuts. We discuss in particular
the potential of the projected International Linear Collider to discover lepton
flavour violation in this class of scenarios, and we compare the estimated
sensitivity with the constraints stemming from the non-observation of rare
decays.
