We prove existence and uniqueness of a mild solution of a stochastic evolution equation driven by a standard $α$-stable cylindrical Lévy process defined on a Hilbert space for $α\in (1,2)$. The coefficients are assumed to map between certain domains of fractional powers of the generator present in the equation. The solution is constructed as a weak limit of the Picard iteration using tightness arguments. Existence of strong solution is obtained by a general version of the Yamada--Watanabe theorem.