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We derive a functional limit theorem for the partial maxima process based ona long memory stationary $\alpha$-stable process. The length of memory in thestable process is parameterized by a certain ergodic-theoretical parameter inan integral representation of the process. The limiting process is no longer aclassical extremal Fr\'{e}chet process. It is a self-similar process with$\alpha$-Fr\'{e}chet marginals, and it has stationary max-increments, aproperty which we introduce in this paper. The functional limit theorem isestablished in the space $D[0,\infty)$ equipped with the Skorohod$M_1$-topology; in certain special cases the topology can be strengthened tothe Skorohod $J_1$-topology.

Maxima of long memory stationary symmetric $\alpha$-stable processes, and self-similar processes with stationary max-increments