Let $A \to S$ be an abelian scheme over an irreducible curve $S$ over complex numbers of relative dimension $g \geq 1$. Let $s: S \to A$ be a non-constant section and denote by $X:=s(S)$. Is it true that $\mathbb{Z}\cdot X:= \cup_{n \in \mathbb{Z}} nX$ is Zariski dense in A?

I expect it is false in general but cannot find a counterexample. Thanks for any guidance.

Let $A \to S$ be an abelian scheme over an irreducible curve $S$ over complex numbers of relative dimension $g \geq 1$. Let $s: S \to A$ be a non-constant section and denote by $X:=s(S)$. Is it true that $\mathbb{Z}\cdot X:= \bigcup_{n \in \mathbb{Z}} nX$ is Zariski dense in A?

I expect it is false in general but cannot find a counterexample. Thanks for any guidance.