Is there an abelian variety $A$ over a field $k$, such that $A(k^{\rm sep})$ is not a divisible group?

The motivation of my question is the following : if $L$ is any algebraically closed field, then $A(L)$ is a divisible group, that is the multiplication map $ [n] : A(L) \to A(L)$ is a surjective group morphism, for every $n \neq 0$ (see corollary 5.10). But if $L$ is only separably closed (e.g. $L = \Bbb F_p(T)^{\rm sep}$), it might not be true anymore ; however I have no counterexample.

If one can find a one-dimensional abelian variety $A$ such that $A(k^{\rm sep})$ is not divisible (in particular, $A$ is an elliptic curve), then I don't understand how one gets the surjective map $[n] : A(k^{\rm sep}) \to A(k^{\rm sep})$ at the beginning of the section 2 here).

[My ultimate goal is to get a long exact sequence in Galois cohomology for abelian varities over $\Bbb F_p(T)$.]


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