  We prove existence and uniqueness of a mild solution of a stochastic
evolution equation driven by a standard $\alpha$-stable cylindrical L\'evy
process defined on a Hilbert space for $\alpha \in (1,2)$. 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.
