  We develop the notion of a geometric covering of a rigid space X, which
yields a much larger class of covering spaces than that studied previously by
de Jong. Geometric coverings of X are closed under disjoint unions and are
\'etale local on X. If X is connected, its geometric coverings form a tame
infinite Galois category, and hence are classified by a topological group. The
definition is based on the property of lifting of "geometric arcs," making it
similar to geometric coverings of schemes studied by Bhatt and Scholze as well
as semicoverings of topological spaces introduced by Brazas.
