# Surjectivity of multiplication by $n$ on elliptic curves

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 $$\Bbb F_p(T)$$.]