Background Facilitating factors and barriers to breast milk feeding (BMF) very preterm (VP) infants have been widely studied at the individual level. We aimed to describe and analyse factors associated with BMF at discharge for VP infants, with a special focus on unit policies aiming to support BMF. Methods We described BMF at discharge in 3108 VP infants enrolled in EPIPAGE‐2, a French national cohort. Variables of interest were kangaroo care during the 1st week of life (KC); unit's policies supporting BMF initiation (BMF information systematically given to mothers hospitalised for threatened preterm delivery and breast milk expression proposed within 6 hours after birth) and BMF maintenance (availability of protocols for BMF and a special room for mothers to pump milk); the presence in units of a professional trained in human lactation and regional BMF initiation rates in the general population. Associations were investigated by multilevel logistic regression analysis, with adjustment on individual factors. Results In total, 47.2% of VP infants received BMF at discharge (range among units 21.1%‐84.0%). Unit policies partly explained this variation, regardless of individual factors. BMF at discharge was associated with KC (adjusted odds ratio (aOR) 2.26 (95% confidence interval (CI) 1.40, 3.65)), with policies supporting BMF initiation (aOR 2.19 (95% CI 1.27, 3.77)) and maintenance (aOR 2.03 (95% CI 1.17, 3.55)), but not with BMF initiation rates in the general population. Conclusion Adopting policies of higher performing units could be an effective strategy for increasing BMF rates at discharge among VP infants.