  We construct a non-sheafy uniform Banach algebra such that a rational
localisation of the Berkovich spectrum does not preserve the uniformity. We
also construct uniform affinoid rings in the sense of Roland Huber such that
rational localisations of the adic spectra do not preserve the uniformity. One
of them is an example of a non-sheafy uniform affinoid ring. We introduce the
notion of local uniformity instead, and prove that the local uniformity implies
the sheaf condition.
