  We study star-finite coverings of infinite-dimensional normed spaces. A
family of sets is called star-finite if each of its members intersects only
finitely many other members of the family. It follows by our results that an
LUR or a uniformly Fr\'echet smooth infinite-dimensional Banach space does not
admit star-finite coverings by closed balls. On the other hand, we present a
quite involved construction proving existence of a star-finite covering of
$c_0(\Gamma)$ by Fr\'echet smooth centrally symmetric bounded convex bodies. A
similar but simpler construction shows that every normed space of countable
dimension (and hence incomplete) has a star-finite covering by closed balls.
