Let $R$ be the ring of integers in a (complete) algebraic closure of $\mathbb Q_p$ with maximal ideal $\mathfrak p$. Suppose I have an Abelian surface $\mathcal A/R$ such that over every $R/\mathfrak p^n$, there exist elliptic curves $E_n, E_n'$ over $R$ with $\mathcal A$ isogenous to $E_n\times E_n'$ over $R/\mathfrak p^n$.

Does this imply that $\mathcal A$ splits over $R$? Note that I do not require any compatibility conditions between the elliptic curves or the isogenies over varying $n$.