  We study subsets of the $n$-dimensional vector space over the finite field
$\mathbb{F}_q$, for odd $q$, which contain either a sphere for each radius or a
sphere for each first coordinate of the center. We call such sets radii
spherical Kakeya sets and center spherical Kakeya sets, respectively.
  For $n\ge 4$ we prove a general lower bound on the size of any set containing
$q-1$ different spheres which applies to both kinds of spherical Kakeya sets.
We provide constructions which meet the main terms of this lower bound.
  We also give a construction showing that we cannot get a lower bound of order
of magnitude~$q^n$ if we take lower dimensional objects such as circles in
$\mathbb{F}_q^3$ instead of spheres, showing that there are significant
differences to the line Kakeya problem.
  Finally, we study the case of dimension $n=1$ which is different and
equivalent to the study of sum and difference sets that cover $\mathbb{F}_q$.
