The descriptive complexity of the set of all closed zero-dimensional subsets of a Polish space

Given a space $X$ we investigate the descriptive complexity class ${\text{G}}_X$ of the set ${\text{FF}}_0(X)$ of all its closed zero-dimensional subsets, viewed as a subset of the hyperspace $\text{FF}(X)$ of all closed subsets of $X$. We prove that $\max \{{\text{G}}_X;X\text{ analytic }\}=\text{pca}$ and $\sup \{{\text{G}}_X;X\text{ Borel }\text{borm}\xi \}\supseteq \mathrm{⅁}\text{bora}\xi$ for any countable ordinal $\xi \ge 1$. In particular we prove that there exists a one-dimensional Polish subpace of $2^{\text{wo}}\times {\mathbb{R}}^2$ for which ${\text{FF}}_0(X)$ is not in the smallest non trivial pointclass closed under complementation and the Souslin operation $\mathcal{A}$.