Untestable assumptions about the association between survival and censoring times can affect the validity of estimates of the survival distribution including the Kaplan–Meier (KM) nonparametric maximum likelihood estimate (MLE). This paper explores the sensitivity of the KM curve to nonignorable censoring by extending the index of local sensitivity to nonignorability (ISNI; Troxel et al., Statistica Sinica, 14, 1221–1237, 2004; Zhang and Heitjan, Biometrics, 62, 1260–1268, 2006) to the case of a nonparametric survival model. The method involves, first, specifying a coarse-data selection model to describe the association between the failure and censoring processes and then evaluating the slope of the nonparametric survival MLE ordinate with respect to a nonignorability parameter in the neighborhood of the ignorable model. We define the nonparametric MLE of the survival curve for a fixed value of the nonignorability parameter and show in a simulation that ISNI analysis effectively captures local sensitivity to nonignorability. The method measures sensitivity in the sense of identifying functionals of the nonparametric MLE that nonignorability, if present, can affect substantially. We demonstrate the method with an application to a trial comparing mechanical assistance to optimal medical management in the treatment of end-stage heart failure. Copyright © 2012 John Wiley & Sons, Ltd.