Keyphrase extraction is a fundamental task in natural language processing. However, existing unsupervised prompt-based methods for Large Language Models (LLMs) often rely on single-stage inference pipelines with uniform prompting, regardless of document length or LLM backbone. Such one-size-fits-all designs hinder the full exploitation of LLMs' reasoning and generation capabilities, especially given the complexity of keyphrase extraction across diverse scenarios. To address these challenges, we propose MAPEX, the first framework that introduces multi-agent collaboration into keyphrase extraction. MAPEX coordinates LLM-based agents through modules for expert recruitment, candidate extraction, topic guidance, knowledge augmentation, and post-processing. A dual-path strategy dynamically adapts to document length: knowledge-driven extraction for short texts and topic-guided extraction for long texts. Extensive experiments on six benchmark datasets across three different LLMs demonstrate its strong generalization and universality, outperforming the state-of-the-art unsupervised method by 2.44% and standard LLM baselines by 4.01% in F1@5 on average. Code is available at https://github.com/NKU-LITI/MAPEX.