The process $(\int_0^t e^{2b_s-b_t}\, ds\ ;\ t\ge 0)$, where $b$ is a real Brownian motion, is known as the geometric 2M-X Matsumoto--Yor process. Remarkably, it enjoys the Markov property. We provide a generalization of this process in the context of Jordan algebras, and we prove the Markov property for this generalization. Our Markov process occurs as a limit of discrete-time AX+B Markov chains on the cone of squares whose invariant probability measures classically yield a Dufresne-type identity for a perpetuity. In particular, the paper provides a generalization to any symmetric cone of the matrix--valued generalization of the Matsumoto--Yor process and Dufresne identity initially developed by Rider--Valkó.