In the paper we study random stable countable zonotopes in R^d, given by [Formula presented], where ⊕ stands for Minkowski sum, Γ_k=∑₁ ^kτ_j, {τ_j,j≥1} are i.i.d. random variables with common standard exponential distribution, and {ε_k,k≥1} are i.i.d. random vectors in R^d with common distribution σ, concentrated on the unit sphere of R^d. The sequences (τ_k) and (ε_k) are supposed to be independent. We consider these random sets as elements of the space K_d of all compact convex subsets of R^d with the Hausdorff distance. The boundary of Z_α has non-trivial Cantor-type structure, and, in the case d=2 and under mild condition on σ, we prove that the Hausdorff dimension of the set of extremal points of the boundary of Z_α is equal to α. As a by-product we provide some kind of geometrical characterization of (non-random) countable zonotopes in the class of zonoids, which are limits of zonotopes.