The Lamperti--Kiu transformation for real-valued self-similar Markov processes (rssMp) states that, associated to each rssMp via a space-time transformation, there is a Markov additive process (MAP). In the case that the rssMp is taken to be an $α$-stable process with $α\in(0,2)$, Chaumont et al. (2013) and Kuznetsov et al. (2014) have computed explicitly the characteristics of the matrix exponent of the semi-group of the embedded MAP, which we henceforth refer to as the {\it Lamperti-stable MAP}. Specifically, the matrix exponent of the Lamperti-stable MAP's transition semi-group can be written in a compact form using only gamma functions. Just as with Lévy processes, there exists a factorisation of the (matrix) exponents of MAPs, with each of the two factors uniquely characterising the ascending and descending ladder processes, which themselves are again MAPs. To the author's knowledge, not a single example of such a factorisation currently exists in the literature. In this article we provide a completely explicit Wiener--Hopf factorisation for the Lamperti-stable MAP. As a consequence of our methodology, we also get additional new results concerning space-time invariance properties of stable processes. Accordingly we develop some new fluctuation identities therewith.