Let $A,B$ be positive dimensional Abelian varieties over a finite field and $p$ be an arbritrary prime. By Zarhin, *Homomorphisms of abelian varieties over finite fields* http://www.math.nyu.edu/~tschinke/books/finite-fields/final/10_zarhin.pdf, Theorem 10.2, one has an isomorphism $$\mathrm{Hom}(A,B) \otimes \mathbf{Z}_p \to \mathrm{Hom}(A(p),B(p))$$ with $A(p)$, $B(p)$ the $p$-divisible groups of $A$ and $B$ (a generalisation of Tate's *Endomorphisms of Abelian Varieties over Finite Fields* to $p = \mathrm{char}(K)$).

Is the Galois action on the $p$-adic Tate module also semisimple as it is for $\ell \neq p$?