Although a large number of provably secure cryptographic primitives have been proposed in the literature, many of these schemes might be broken in practice because of various leakage attacks. Therefore, the leakage resilience property should be considered in designing these primitives. However, in identity-based cryptography, most of the existing leakage-resilient identity-based encryption (IBE) schemes suffer some limitations: they either resist the leakage attacks in the selective identity security model or achieve the chosen-ciphertext attack (CCA) security based on a non-static assumption. To further solve the above problems, in this paper, an IBE scheme with adaptive leakage-resilient chosen-plaintext attacks (CPA) security is proposed, and its security is rigorously proved in the random oracle model under a classic static assumption, e.g., decisional bilinear Diffie-Hellman (DBDH) assumption. In addition, the leakage-resilient CCA security can be obtained from the above CPA secure IBE scheme. That is, a generic method created leakage-resilient CCA secure public-key encryption is proposed, and the leakage-resilient CCA secure IBE scheme can also be created. Therefore, we propose a noval method to create leakage-resilient IBE scheme with CCA security from static complexity assumption.