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 ∈ (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.