Generic expansion of an abelian variety by a subgroup

Let $A$ be an abelian variety in a field of characteristic $0$. We prove that the expansion of $A$ by a generic divisible subgroup of $A$ with the same torsion exists provided $A$ has few algebraic endomorphisms, namely $\mathrm{End}(A)=\mathbb Z$. The resulting theory is $\mathrm{NSOP}_1$ and not simple. Note that there exist abelian varieties $A$ with $\mathrm{End}(A) = \mathbb{Z}$ of any genus. We indicate how this result can be extended to any simple abelian variety by considering the expansion by a predicate for some submodule over $\mathrm{End}(A)$.