Let $\cal A$ be a smooth commutative group scheme over $S$, where $S$ is the spectrum of a discrete valuation ring with fraction field $K$ and residue field $k$. Suppose that $A:={\cal A}_K$ is an abelian variety and that ${\cal A}_k$ is a semiabelian variety. Let $\cal N$ be the Néron model of $A$ over $S$. It is then standard that the connected component of $\cal N$ is isomorphic to $\cal A$. Let $P\in A(K)$. Then, by the definition of the Néron model, $P$ extends to a section $\widetilde P\in{\cal N}(S)$. Since the group of components of ${\cal N}_k$ is finite, there is thus a natural number $n$ such that $n\cdot \widetilde P\in{\cal A}(S)$. In particular, $n\cdot P\in A(K)$ extends to an element of ${\cal A}(S)$.
My question is : is there an elementary proof of the fact that for any $P\in A(K)$, there exists a natural number $n$ such that $n\cdot P\in A(K)$ extends to an element of ${\cal A}(S)$ ? By "elementary", I mean a proof which does not use the existence of Néron models (and also, ideally, not partial rigid analytic uniformisations).
