Wondering why Andrew D King's answer (or Konrad Swanepoel's comment on it) was not upvoted. (As a newbie, I cannot, nor can I comment.) If we cheekily let $\hat{\kappa}_n$ denote the hemispherical kissing number in $n$-dimensional space, defined here to mean the maximum number of mutually disjoint $n$-dimensional unit hyperspheres tangentially adjacent to a $n$-dimensional unit hemihypersphere, then the corresponding chromatic number is at most $\hat{\kappa}_n+1$ (by Andrew's argument). Konrad mentioned that $\hat{\kappa}_3\le8$, giving an upper bound of $9$ for colouring sphere packings. For larger $n$, certainly the base in the Kabatiansky-Levenshtein bound on $\kappa_n$ can be beaten for $\hat{\kappa}_n$!
Edit: My original post would have implied a nice short proof of the $5$-colour theorem, but alas there are easy counterexamples to the claim that the maximum minimum degree (a.k.a. degeneracy) is at most $\hat{\kappa}_n$.
However, $\kappa_n$ is an upper bound for colouring $n$-dimensional hypersphere packings using Andrew's argument.
Wondering why Andrew D King's answer (or Konrad Swanepoel's comment on it) was not upvoted. (As a newbie, I cannot, nor can I comment.) If we cheekily let $\hat{\kappa}_n$ denote the hemispherical kissing number in $n$-dimensional space, defined here to mean the maximum number of mutually disjoint $n$-dimensional unit hyperspheres tangentially adjacent to a $n$-dimensional unit hemihypersphere, then the corresponding chromatic number is at most $\hat{\kappa}_n+1$ (by Andrew's argument). Konrad mentioned that $\hat{\kappa}_3\le8$, giving an upper bound of $9$ for colouring sphere packings. For larger $n$, certainly the base in the Kabatiansky-Levenshtein bound on $\kappa_n$ can be beaten for $\hat{\kappa}_n$!