  We show the pathwise uniqueness for stochastic partial differential equation
driven by a cylindrical $\alpha$-stable process with H\"older continuous drift,
thus obtaining an infinite dimensional generalization of the result of Priola
[Osaka J. Math., 2012] in the case $H=\mathbb{R}^d$. The proof is based on an
infinite dimensional Kolmogorov equation with non-local operator.
