Designing optimal treatment plans for patients with comorbidities requires accurate cause-specific mortality prognosis. Motivated by the recent availability of linked electronic health records, we develop a nonparametric Bayesian model for survival analysis with competing risks, which can be used for jointly assessing a patient's risk of multiple (competing) adverse outcomes. The model views a patient's survival times with respect to the competing risks as the outputs of a deep multi-task Gaussian process (DMGP), the inputs to which are the patients' covariates. Unlike parametric survival analysis methods based on Cox and Weibull models, our model uses DMGPs to capture complex non-linear interactions between the patients' covariates and cause-specific survival times, thereby learning flexible patient-specific and cause-specific survival curves, all in a data-driven fashion without explicit parametric assumptions on the hazard rates. We propose a variational inference algorithm that is capable of learning the model parameters from time-to-event data while handling right censoring. Experiments on synthetic and real data show that our model outperforms the state-of-the-art survival models.