Summary: This article deals with the semiparametric analysis of multivariate survival data with random block (group) effects. Survival times within the same group are correlated as a consequence of a frailty random block effect. The standard approaches assume either a parametric or a completely unknown baseline hazard function. This paper considers an intermediate solution, that is, a nonparametric function that is reasonably smooth. This is accomplished by a Bayesian model in which the conditional proportional hazards model is used with a correlated prior process for the baseline hazard. The posterior likelihood based on data, as well as the prior process, is similar to the discretized penalized likelihood for the frailty model. The methodology is exemplified with the recurrent kidney infections data of C. A. McGilchrist and C. W. Aisbett [Biometrics 47, 461–466 (1991)], where the times to infections within the same patients are expected to be correlated. The reanalysis of the data has shown that the estimates of the parameters of interest and the associated standard errors depend on the prior knowledge about the smoothness of the baseline hazard.