Variants of the de Jong fundamental group

abstract: For a rigid space $X$, we answer two questions of de~Jong about the category $\mathbf{Cov}^\mathrm{adm}_X$ of coverings which are locally in the admissible topology on $X$ the disjoint union of finite \'etale coverings: we show that this class is different from the one used by de~Jong, but still gives a tame infinite Galois category. In addition, we prove that the objects of $\mathbf{Cov}^{\text{\'et}}_X$ (with the analogous definition) correspond precisely to locally constant sheaves for the pro-\'etale topology defined by Scholze.