Very recently, side-channel attacks have threatened all traditional cryptographic schemes. Typically, in traditional cryptography, private/secret keys are assumed to be completely hidden to adversaries. However, by side-channel attacks, an adversary may extract fractional content of these private/secret keys. To resist side-channel attacks, leakage-resilient cryptography is a countermeasure. Identity-based public-key system (ID-PKS) is an attractive public-key setting. ID-PKS settings not only discard the certificate requirement, but also remove the construction of the public-key infrastructure. For solving the user revocation problem in ID-PKS settings, revocable ID-PKS (RID-PKS) setting has attracted significant attention. Numerous cryptographic schemes based on RID-PKS settings have been proposed. However, under RID-PKS settings, no leakage-resilient signature or encryption scheme is proposed. In this article, we present the first leakage-resilient revocable ID-based signature (LR-RIBS) scheme with cloud revocation authority (CRA) under the continual leakage model. Also, a new adversary model of LR-RIBS schemes with CRA is defined. Under this new adversary model, security analysis is made to demonstrate that our LR-RIBS scheme with CRA is provably secure in the generic bilinear group (GBG) model. Finally, performance analysis is made to demonstrate that our scheme is suitable for mobile devices.