In the presence of right-censored data with covariates, the conditional Kaplan-Meier estimator (also known as the Beran estimator) consistently estimates the conditional survival function of the random follow-up for the event of interest. However, a necessary condition is the unambiguous knowledge of whether each individual is censored or not, which may be incomplete in practice. We therefore propose a study of the Beran estimator when the censoring indicators are generic random variables and discuss necessary conditions for the efficiency of the Beran estimator. From this, we provide a new estimator for the conditional survival function with missing not at random (MNAR) censoring indicators based on a conditional copula model for the missingness mechanism. In addition to the theoretical results, we illustrate how the estimators work for small samples through a simulation study and show their practical applicability by analyzing synthetic and real data.