  We study spherical completeness of ball spaces and its stability under
expansions. We give some criteria for ball spaces that guarantee that spherical
completeness is preserved when the ball space is closed under unions of chains.
This applies in particular to the spaces of closed ultrametric balls in
ultrametric spaces with linearly ordered value sets, or more generally, with
countable narrow value sets. We show that in general, chain union closures of
ultrametric spaces with partially ordered value sets do not preserve spherical
completeness. Further, we introduce and study the notions of chain union
stability and of chain union rank, which measure how often the process of
closing a ball space under all unions of chains has to be iterated until a ball
space is obtained that is closed under unions of chains.
