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Many popular survival models rely on restrictive parametric, or semiparametric, assumptions that could provide erroneous predictions when the effects of covariates are complex. Modern advances in computational hardware have led to an increasing interest in flexible Bayesian nonparametric methods for time‐to‐event data such as Bayesian additive regression trees (BART). We propose a novel approach that we call nonparametric failure time (NFT) BART in order to increase the flexibility beyond accelerated failure time (AFT) and proportional hazard models. NFT BART has three key features: (1) a BART prior for the mean function of the event time logarithm; (2) a heteroskedastic BART prior to deduce a covariate‐dependent variance function; and (3) a flexible nonparametric error distribution using Dirichlet process mixtures (DPM). Our proposed approach widens the scope of hazard shapes including nonproportional hazards, can be scaled up to large sample sizes, naturally provides estimates of uncertainty via the posterior and can be seamlessly employed for variable selection. We provide convenient, user‐friendly, computer software that is freely available as a reference implementation. Simulations demonstrate that NFT BART maintains excellent performance for survival prediction especially when AFT assumptions are violated by heteroskedasticity. We illustrate the proposed approach on a study examining predictors for mortality risk in patients undergoing hematopoietic stem cell transplant (HSCT) for blood‐borne cancer, where heteroskedasticity and nonproportional hazards are likely present. [ABSTRACT FROM AUTHOR]