Key exposure is a major threat to secure cryptosystems. To mitigate the impact caused by key-compromise attacks, a key-insulated cryptographic mechanism is a better alternative. For securing the large message communication in peer-to-peer networks, in this paper, we propose the first novel identity-based key-insulated encryption (IB-KIE) scheme with message linkages. Our scheme has the properties of unbounded time periods and random-access key-updates. In the proposed scheme, each client can periodically update his private key while the corresponding public one remains unchanged. The essential security assumption of our proposed scheme is based on the well-known bilinear Diffie-Hellman problem (BDHP). To ensure the practical feasibility, we also formally prove that the proposed scheme achieves the security requirement of confidentiality against indistinguishability under adaptive chosen-ciphertext attacks (IND-CCA2) in the random oracle model.