Safe navigation of drones in the presence of adversarial physical attacks from multiple pursuers is a challenging task. This paper proposes a novel approach, asynchronous multi-stage deep reinforcement learning (AMS-DRL), to train adversarial neural networks that can learn from the actions of multiple evolved pursuers and adapt quickly to their behavior, enabling the drone to avoid attacks and reach its target. Specifically, AMS-DRL evolves adversarial agents in a pursuit-evasion game where the pursuers and the evader are asynchronously trained in a bipartite graph way during multiple stages. Our approach guarantees convergence by ensuring Nash equilibrium among agents from the game-theory analysis. We evaluate our method in extensive simulations and show that it outperforms baselines with higher navigation success rates. We also analyze how parameters such as the relative maximum speed affect navigation performance. Furthermore, we have conducted physical experiments and validated the effectiveness of the trained policies in real-time flights. A success rate heatmap is introduced to elucidate how spatial geometry influences navigation outcomes. Project website: https://github.com/NTU-ICG/AMS-DRL-for-Pursuit-Evasion.