Galois invariants of the Tate module of an elliptic curve over a number field

We’re looking in the local, complete situation above $\ell$ at the $p^m$-torsion points of $E$ for all $m$. What does it mean to say that $T_p(E)^{G_v}\ne0$, where $G_v=G_{K_v}$, the Galois group of an algebraic closure of $K_v$ over $K_v$? It would mean that there was a consistent sequence of $p^m$-torsion points of $E$, in particularly infinitely many of them, that are rational over a finite extension of $K_v$. But since we have the exact sequence $$ 0\>\rightarrow\>\widehat E(\mathfrak m)\>\rightarrow E(\mathfrak o)\>\rightarrow\>\tilde E(\kappa)\rightarrow\>0\,, $$ this can’t happen. Here, $\widehat E(\mathfrak m)$ is the points of the formal group of $E$ with values in the maximal ideal $\mathfrak m$ of the ring $\mathfrak o$ of $v$-integers of $K_v$ (or some finite extension if necessary); $E(\mathfrak o)$ is the $\mathfrak o$-points of $E$ (same as the $K_v$-points), and $\tilde E(\kappa)$ is the group of points of the reduced curve $\tilde E$ rational over the finite field $\kappa$ of characteristic $\ell$. But the points of the formal group are uniquely divisible by any prime different from $\ell$, so there’s no $p$-torsion there; and there are only finitely many points over the finite field. So no good.