<p>In multi-arm Phase II oncology trials, survival endpoints require prolonged follow-up, limiting the practicality of outcome-adaptive randomization (OAR). We propose a novel Bayesian adaptive randomization (BAR) design for many-to-one comparisons that leverages short-term surrogate responses to guide allocation while retaining progression-free survival (PFS) as the primary endpoint. Building on Huang et al. (2009), our key contribution is a unified framework that links multinomial response categories to arm-specific Weibull survival distributions via a Bayesian mixture model. The proposed BAR design enables continually posterior updating of mean survival and interim futility monitoring such that final selection can be made based on the posterior probability that an experimental arm exceeds control by a prespecified clinically meaningful margin. We examine two adaptive allocation strategies: one driven by transformed posterior superiority probabilities and another that targets desired allocations while correcting observed imbalances. Simulation studies across realistic scenarios demonstrate nominal 5% Type I error rate control, high power under heterogeneous treatment effects, and robust performance under survival-model misspecification and alternative prior specifications.</p>