Let $(S,d)$ be a discrete space endowed with the metric $d$, $d(x,y)=0$ if $x=y$ and $d(x,y)=1$ if $x\neq y$. It is well-known that the metrizable space space $S^{\omega_0}$ obtained by doing the Cartesian product of $\omega_0$-many copies of $(S,d)$, endowed with the usual product metric, is strongly zero-dimensional. A strongly zero-dimensional (Tychonoff) space is a space satisfying that every finite co-zero cover has a refinement which is a finite partition of clopen sets (see for instance, Engelking's General Topolpogy Section 6.2 and Example 7.3.14).
Now, for $\kappa >\omega_0$, is $S^\kappa$, endowed with the usual product topology, strongly zero-dimensional?