Trustworthy biomedical question answering (QA) systems must not only provide accurate answers but also justify them with current, verifiable evidence. Retrieval-augmented approaches partially address this gap but lack mechanisms to iteratively refine poor queries, whereas self-reflection methods kick in only after full retrieval is completed. In this context, we introduce PubMed Reasoner, a biomedical QA agent composed of three stages: self-critic query refinement evaluates MeSH terms for coverage, alignment, and redundancy to enhance PubMed queries based on partial (metadata) retrieval; reflective retrieval processes articles in batches until sufficient evidence is gathered; and evidence-grounded response generation produces answers with explicit citations. PubMed Reasoner with a GPT-4o backbone achieves 78.32% accuracy on PubMedQA, slightly surpassing human experts, and showing consistent gains on MMLU Clinical Knowledge. Moreover, LLM-as-judge evaluations prefer our responses across: reasoning soundness, evidence grounding, clinical relevance, and trustworthiness. By orchestrating retrieval-first reasoning over authoritative sources, our approach provides practical assistance to clinicians and biomedical researchers while controlling compute and token costs.