Do integrable mappings have the Painlevé property?

We present an integrability criterion for discrete-time systems that is the equivalent of the Painlev\'e property for systems of a continuous variable. It is based on the observation that for integrable mappings the singularities that may appear are confined, i. e. , they do not propagate indefinitely when one iterates the mapping. Using this novel criterion we show that there exists a family of nonautonomous integrable mappings which includes the discrete Painlev\'e equations by P₈, recently derived in a model of two-dimensional quantum gravity, and P₈₈, obtained as a similarity reduction of the integrable modified Korteweg--de Vries lattice. These systems possess Lax pairs, a well-known integrability feature.