We study survival analysis with positive–unlabelled event-time data, where censoring times and covariates are observed for all subjects but event times only for labelled positives. Under conditionally independent censoring and allowing the labelling probability to depend on censoring time and covariates, conditioning on these variables eliminates the unknown labelling probability from the event-time density of labelled positives. We show that this conditional density identifies the reverse hazard and the relative event-time distribution over the observed follow-up range, while absolute event probabilities require additional information or restrictions. We develop parametric and semiparametric estimators based on the conditional likelihood and establish their large-sample properties. In regular parametric models, the estimator is semiparametrically efficient when the labelling mechanism is locally unrestricted. We also show that information about scale parameters can vanish under limited follow-up and propose a sensitivity analysis for event-timedependent labelling. In simulations, we examine finite-sample performance under varying follow-up and label selection. In an empirical application, we study the time from clinical-trial completion to the first results disclosure recorded through a registry posting or qualifying journal publication.