A famous result of V.~Klee from 1981 is that the Banach space $\ell_1(\kappa)$ admits a disjoint tiling by balls of radius $1$, for all cardinals $\kappa$ with $\kappa^\omega =\kappa$. Klee also observed that the smallest cardinal in which such a tiling might exist is $\kappa= 2^{\aleph_0}$, leaving open the question whether, for $\kappa< 2^{\aleph_0}$, $\ell_1(\kappa)$ might admit a tiling by balls at all. In the talk we shall answer this question and we will also present some classes of Banach spaces that `almost' admit a tiling by balls of radius 1. The talk will be based on (parts of) two papers, joint with Carlo Alberto De Bernardi, \c{S}eyda Sezgek, and Jacopo Somaglia.