According to a talk that I found on the web, it is a theorem of Voronoi that every indecomposable root lattice is extreme. Also the $E_8$ lattice is the union of two copies of the $D_8$ lattice with the same sphere radius. And, of course, the rotational symmetry group of the $E_8$ is transitive on roots. So I think that that gives it to you: If you put two small, flat dents in a round ball in $\mathbb{R}^8$, then you cannot deform the $E_8$ lattice packing, because there is a $D_8$ lattice inside that is far away from the dents.
It looks like the
The same argument works in 24 dimensionsfor the Leech lattice, which contains a $D_{24}$ sublattice of index 8192. It also works for $E_7$, because it contains $A_7$. (Also $E_8$ contains $A_8$, but not in the same way, since $[E_8:A_8] = 3$ while $[E_7:A_7] = 2$.)
However, I do not think that it is known, nor even a conjecture with strong evidence, that the kissing number of the best lattice sphere packing in dimension $n \to \infty$ is more than the bare minimum $n(n+1)/2$.n(n+1)$.
