On cohomology of locally profinite sets

We construct a locally profinite set of cardinality \(\aleph_{\omega}\) with infinitely many first cohomology classes of which any distinct finite product does not vanish. Building on this, we construct the first example of a nondescendable faithfully flat map between commutative rings of cardinality \(\aleph_{\omega}\) within Zermelo--Fraenkel set theory.