Abstract Insular woodiness refers to the evolutionary transition from herbaceousness towards derived woodiness on (sub)tropical islands and leads to island floras that have a higher proportion of woody species compared to floras of nearby continents. Several hypotheses have tried to explain insular woodiness since Darwin's original observations, but experimental evidence why plants became woody on islands is scarce at best. Here, we combine experimental measurements of hydraulic failure in stems (as a proxy for drought stress resistance) with stem anatomical observations in the daisy lineage (Asteraceae), including insular woody Argyranthemum species from the Canary Islands and their herbaceous continental relatives. Our results show that stems of insular woody daisies are more resistant to drought‐induced hydraulic failure than the stems of their herbaceous counterparts. The anatomical character that best predicts variation in embolism resistance is intervessel pit membrane thickness ( T PM ), which can be functionally linked with air bubble dynamics throughout the 3D vessel network. There is also a strong link between T PM vs. degree of woodiness and thickness of the xylem fibre wall vs. embolism resistance, resulting in an indirect link between lignification and resistance to embolism formation. Thicker intervessel pit membranes in Argyranthemum functionally explain why this insular woody genus is more embolism resistant to drought‐induced failure compared to the herbaceous relatives from which it has evolved, but additional data are needed to confirm that palaeoclimatic drought conditions have triggered wood formation in this daisy lineage. A plain language summary is available for this article.< Réduire