We describe the solution of an optimal stopping problem for a stable Lévy process killed at state-dependent rate. Since the killing occurs only when the process is below zero, this can be regarded as a model for bankruptcy when the process represents a capital level. The killing rate is chosen in such a way that the killed process remains self-similar, and the solution to the optimal stopping problem is obtained by characterising a self-similar Markov process associated with the stable process. The optimal stopping strategy is to stop upon first passage into an interval, found explicitly in terms of the parameters of the model.