Dendrons and their products admit a natural, continuous median operator. We prove that there exists a two‐dimensional metric continuum with a continuous median operator, for which there is no median‐preserving embedding in a product of finitely many dendrons. Our method involves ideas and results concerning graph colouring and abstract convexity. The main result answers a question in [16] negatively, and is sharply contrasting with a result of Stralka [15]on embeddings of compact lattices.