In this study, we explore the robustness of cooperative multi-agent reinforcement learning (c-MARL) against Byzantine failures, where any agent can enact arbitrary, worst-case actions due to malfunction or adversarial attack. To address the uncertainty that any agent can be adversarial, we propose a Bayesian Adversarial Robust Dec-POMDP (BARDec-POMDP) framework, which views Byzantine adversaries as nature-dictated types, represented by a separate transition. This allows agents to learn policies grounded on their posterior beliefs about the type of other agents, fostering collaboration with identified allies and minimizing vulnerability to adversarial manipulation. We define the optimal solution to the BARDec-POMDP as an ex interim robust Markov perfect Bayesian equilibrium, which we proof to exist and the corresponding policy weakly dominates previous approaches as time goes to infinity. To realize this equilibrium, we put forward a two-timescale actor-critic algorithm with almost sure convergence under specific conditions. Experiments on matrix game, Level-based Foraging and StarCraft II indicate that, our method successfully acquires intricate micromanagement skills and adaptively aligns with allies under worst-case perturbations, showing resilience against non-oblivious adversaries, random allies, observation-based attacks, and transfer-based attacks.