Individuals often respond differently to identical treatments, and characterizing such variability in treatment response is an important aim in the practice of personalized medicine. To this end, we describe a nonparametric accelerated failure time model for analyzing heterogeneous treatment effects (HTE) when patient outcomes are time-to-event. By utilizing Bayesian additive regression trees and a mean-constrained Dirichlet process mixture model, our approach provides a flexible regression function model while placing few restrictions on the baseline hazard. Our method leads to natural estimates of individual treatment effect and has the flexibility to address many major goals of HTE assessment. Moreover, our approach requires little user input in terms of model specification for treatment covariate interactions or for parameter tuning. Our procedure shows strong predictive performance while also exhibiting good frequentist properties in terms of parameter coverage and mitigation of spurious findings of HTE. We illustrate the merits of our proposed approach with a detailed analysis of two large clinical trials for the treatment of congestive heart failure using an ACE inhibitor.