Let $A$ be an abelian variety over a $p$-adic field $K_v$, i.e., $K_v$ is a finite field extension of $\mathbb Q_p$, for $p$ a prime number. Denote by $k_v$ the residue field of $K_v$ and let $\mathcal{A}_v^{0}(k_v)$ be the smooth part of the $k_v$-rational points of the modulo $v$ reduced variety $A$, i.e., the $k_v$-rational points of the connected component of the identity section of the special fiber at $v$ of the Néron model $\mathcal{A}/\mathcal{O}_K$ of $A$. Denote by $A_0(K_v)$ the preimage of the reduction-mod-$v$-map on $\mathcal{A}_v^{0}(k_v)$ and by $A_1(K_v)$ the kernel, hence we have the short exact sequence $$0 \rightarrow A_1(K_v) \rightarrow A_0(K_v) \rightarrow \mathcal{A}_v^{0}(k_v) \rightarrow 0.$$ Why is $A_1(K_v)$ a pro-$p$ group?

(Is there a standard reference for this fact?)