In 2014, a new security definition of a revocable identity-based signature (RIBS) with signing key exposure resistance was introduced. Based on this new definition, many scalable RIBS schemes with signing key exposure resistance were proposed. However, the security of these schemes is based on traditional complexity assumption, which is not secure against attacks in the quantum era. Lattice-based cryptography has many attractive features, and it is believed to be secure against quantum computing attacks. We reviewed existing lattice-based RIBS schemes and found that all these schemes are vulnerable to signing key exposure. Hence, in this paper, we propose the first lattice-based RIBS scheme with signing key exposure resistance by using the left-right lattices and delegation technology. In addition, we employ a complete subtree revocation method to ensure our construction meeting scalability. Finally, we prove that our RIBS scheme is selective-ID existentially unforgeable against chosen message attacks (EUF-sID-CMA) under the standard short integer solutions (SIS) assumption in the random oracle model.