We consider the estimation of measures of persistent poverty in panel surveys with missing data, focusing on the persistent poverty headcount, its duration-adjusted variant, and a related measure used by the European Union as an indicator of the risk of persistent poverty. We develop a partial identification approach to allow for data missing-not-at-random and apply it to panel data from Peru for 2007-11. The “worst case” bounds are very wide, but we achieve much more precise identification by adding a set of weak a priori restrictions. Standard non-response weighting adjustments cannot be relied upon to remove missing-data bias.