Syllogistic reasoning is crucial for Natural Language Inference (NLI). This capability is particularly significant in specialized domains such as biomedicine, where it can support automatic evidence interpretation and scientific discovery. This paper presents SylloBio-NLI, a novel framework that leverages external ontologies to systematically instantiate diverse syllogistic arguments for biomedical NLI. We employ SylloBio-NLI to evaluate Large Language Models (LLMs) on identifying valid conclusions and extracting supporting evidence across 28 syllogistic schemes instantiated with human genome pathways. Extensive experiments reveal that biomedical syllogistic reasoning is particularly challenging for zero-shot LLMs, which achieve an average accuracy between 70% on generalized modus ponens and 23% on disjunctive syllogism. At the same time, we found that few-shot prompting can boost the performance of different LLMs, including Gemma (+14%) and LLama-3 (+43%). However, a deeper analysis shows that both techniques exhibit high sensitivity to superficial lexical variations, highlighting a dependency between reliability, models' architecture, and pre-training regime. Overall, our results indicate that, while in-context examples have the potential to elicit syllogistic reasoning in LLMs, existing models are still far from achieving the robustness and consistency required for safe biomedical NLI applications.