Tilings and coverings by balls in $\ell_1$

A famous result of 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. Our main result answers this question in the negative, proving in particular that $\ell_1$ does not admit any tiling by balls. We also give a companion result about star-$n$-finite coverings by balls of $\ell_1(\kappa)$ and we give a construction of a star-finite tiling of $\mathcal{X} \oplus_\infty c_{00}$, for each space $\mathcal{X}$ whose dimension is at most countable.