This manuscript develops a novel Bayesian parametric proportional hazards regression model for right-censored survival data. The underlying baseline hazard function is modelled via integrated splines that, under certain conditions, guarantee monotonicity. The Bayesian fused lasso prior distribution is employed to control smoothness of the baseline hazard function estimate and to automatically select important covariates. To obtain samples from the posterior distribution, we use Hamiltonian Monte Carlo in conjunction with Proximal MCMC. We demonstrate the usefulness of our methodology on two real data sets.