This paper develops a fully distributed attack-resilient secondary control framework for DC microgrids, in the presence of unknown unbounded attacks on control input channels. The secondary control of each converter consists of an average consensus-based voltage regulator and a consensus-based current regulator that use relative information from neighboring converters. This distributed control manner relies on localized control and a sparse communication network and, therefore, could be prone to malicious attacks that deteriorate the consensus performance and even destabilize the overall microgrids. In contrast to the existing remedies for bounded disturbances/noises, this paper considers unknown attacks that are intentionally unbounded to maximize their damage on the microgrid. A fully distributed adaptive attack-resilient secondary control framework is established for DC microgrids to mitigate the adverse effect of unbounded attacks. Rigorous proofs, based on Lyapunov techniques, show that the proposed method guarantees the uniformly ultimately bounded convergence for both global voltage regulation and proportional load sharing objectives under unbounded attacks. Moreover, the asymptotic stability of the overall closed-loop microgrid system is achieved in the presence of bounded attacks. Experimental results are illustrated in a hardware-in-the-loop environment to validate the effectiveness of the proposed approach.