  We show that any finite family of pairwise intersecting balls in
$\mathbb{E}^n$ can be pierced by $(\sqrt{3/2}+o(1))^n$ points improving the
previously known estimate of $(2+o(1))^n$. As a corollary, this implies that
any $2$-illuminable spiky ball in $\mathbb{E}^n$ can be illuminated by
$(\sqrt{3/2}+o(1))^n$ directions. For the illumination number of convex spiky
balls, i.e., cap bodies, we show an upper bound in terms of the sizes of
certain related spherical codes and coverings. For large dimensions, this
results in an upper bound of $1.19851^n$, which can be compared with the
previous $(\sqrt{2}+o(1))^n$ established only for the centrally symmetric cap
bodies. We also prove the lower bounds of $(\tfrac{2}{\sqrt{3}}-o(1))^n$ for
the three problems above.
