Let $K = \mathbb{F}_q((t))$, and let $A_{/K}$ be a nontrivial abelian variety. Then $A(K)$ is a compact $K$-adic Lie group. What can be said about its structure?

By way of comparison, if $K/\mathbb{Q}_p$ is an extension of degree $d$ and $A$ has dimension $g$, then $p$-adic Lie theory shows that $A(K) \cong \mathbb{Z}_p^{dg} \oplus T$, where $T$ is a finite group. I am looking for a similar description in the positive characteristic case.

I've wondered about this off and on for years but all of a sudden I have a good reason to know: in particular, I would like to know the structure of $A(K)/p^aA(K)$, which I suspect is always an infinite group of exponent $p^a$. This is the case for e.g. $p$-adically uniformized abelian varieties, unless I am very much mistaken.

In particular, in the elliptic curve case it would be nice if the height of the formal group told the majority of the story, in the sense that e.g. if $E_1$ and $E_2$ were any two ordinary elliptic curves over $K$, then $E_1(K)$ and $E_2(K)$ would admit isomorphic finite-index subgroups. Is this true?

**Added**: After seeing Professor Lubin's answer I can be more precise. I would like a proof that $A(K) \cong \left(\prod_{i=1}^{\infty} \mathbb{Z}_p \right) \oplus T$, where $T$ is a finite group.

finite$p$-torsion, a small compact open subgroup $U$ in $A(K)$ has $U[p]=1$, so $U$ is a countable direct product of copies of $\mathbf{Z}_p$. It is a countably infinite product if $A(K)/(p)$ is infinite. Suppose $A(K)/(p)$ is finite, so $p\cdot A(K)$ isopenin $A(K)$. In the formal group of the Neron model $N$ over $R$ with characteristic $p$, $[p]=V_N \circ F_N$. Thus, $[p]^{\ast}(X_j) \in R[[X_1^p,\dots,X_g^p]]$ for all $j$, so openness isimpossibleby measure-theoretic reasons. Thus, $U=\prod_{n=1}^{\infty}\mathbf{Z}_p$. $\endgroup$