Identity-based hash proof system is a basic and important primitive. It is widely utilized to construct cryptographic schemes and protocols that are secure against key-leakage attacks. In this paper, we introduce the concept of updatable identity-based hash proof system, in which the related master secret key and the identity secret key can be updated securely. Then, we instantiate this primitive based on lattices in the standard model. Moreover, we introduce an application of this new primitive by giving a generic construction of leakage-resilient public-key encryption schemes with anonymity. This construction can be considered as the integration of the bounded-retrieval model and the continual leakage model. Compared with the existing leakage-resilient schemes, our construction not only is more efficient but also can resist much more key leakage.