In this paper, we propose a novel class of Piecewise Deterministic Markov Processes (PDMPs) that are designed to sample from probability distributions $π$ supported on a convex set $\mathcal{M}$. This class of PDMPs adapts the concept of a mirror map from convex optimisation to address sampling problems. The corresponding algorithms provide unbiased samples that respect the constraints and, moreover, allow for exact subsampling. We demonstrate the advantages of these algorithms against a range of constrained sampling problems where the proposed algorithms outperform state of the art stochastic differential equation-based methods.