Detailed routing is a crucial challenge in modern integrated circuit (IC) design. Due to the continuous increase in design complexity and complicated design rules, avoiding routing conflicts between nets becomes more and more challenging. Conventional routing strategies like the rip-up and re-route scheme may need to spend huge efforts on avoiding conflicts between nets with overlapping routing areas. To resolve this challenge, in this paper, we propose a detailed router based on multi-agent reinforcement learning for handling conflicting nets. First, we approximate nets of detailed routing as agents and regard the pin-connection task as path planning to achieve the asynchronization of routing. Second, we assign each agent a local field of view to reduce feature size and difficulty in training. Finally, in order to eliminate routing congestion, we set an information storage unit for the information communication of each agent. The evaluation results show that the proposed multi-agent reinforcement learning scheme outperforms the baseline learning methods by 11.6%.