Given an elliptic curve $E_{/\mathbb{Q}}$ (or more generally, a number field) and a prime $p$, there is a standard short exact sequence
$$0\rightarrow E(\mathbb{Q})\otimes \mathbb{Q}_p/\mathbb{Z}_p\rightarrow Sel_{p^\infty}(E/\mathbb{Q})\rightarrow Sha(E/\mathbb{Q})[p^\infty]\rightarrow 0$$, where $Sha(E/\mathbb{Q})$ is the the Tate-Shafarevich group and $Sel_{p^\infty}(E/\mathbb{Q})$ the $p$-primary Selmer group. This Selmer group is a subgroup of $H^1(Gal(\bar{\mathbb{Q}}/\mathbb{Q}), E[p^\infty])$ consisting of classes that localize to classes that are in the image of all local kummer maps. The Selmer group is question is cofinitely generated as a $\mathbb{Z}_p$-module.
Consider a function field analog. Let $F=\mathbb{F}_q(T)$ be the rational function field with $A:=\mathbb{F}_q[T]$ and $\phi$ be a Drinfeld module over $F$ with coefficients in $A$. Let $\mathfrak{p}$ be a non-zero prime of $A$ and $\phi[\mathfrak{p}^\infty]$ its $\mathfrak{p}^\infty$ torsion. Since $\phi$ is of generic $A$-characteristic this is a module over $G_F:=Gal(F^{sep}/F)$. One can study a similar definition for the Selmer group associated to $\phi[mathfrak{p}^\infty]$, however the issue seems to be that it does not seem to be cofinitely generated as an $A_{\mathfrak{p}}$-module. This is because the analog of the Mordell Weil theorem does not hold in this context. Indeed, it was proven by Poonen that the twisted $A$-module $^\phi A$ consists of a direct sum of a finite torsion part and an infinitely generated free $A$-module (a countably infinite direct sum). On the other hand, it seems to me that for all primes $\mathfrak{l}\neq \mathfrak{p}, \infty$ at which $\phi$ has good reduction, the local kummer map should be $0$ thanks to the theory of the formal Drinfeld module. Indeed, there is a submodule of the "local points" of finite index which has no $\mathfrak{p}$-primary part (cf. section 6.5 of "Drinfeld modules"--Papikian).
So it seems to me that the Selmer group in this context consists of cohomology classes that are indeed unramified at all primes of good reduction. So why exactly is this module not cofinitely generated? I'm assuming that perhaps its because there is a lot of wild ramification at $\infty$ (and more controlled wild ramification at $\mathfrak{p}$). On the other hand, if one considers modified "Mordell-Weil" group $e_{\phi}^{-1}(^\phi A)$ defined by Taelman (cf. p.475 in loc. cit.), it turns out to be discrete and co-compact. So I wonder if the required modification to the Selmer group is to impose some sort of strict or unramified condition at $\infty$? Also something like a "Greenberg condition" at $\mathfrak{p}$ would help. And will this then be cofinitely generated and give one relevant information about the arithmetic of the Drinfeld module?
I'm aware that there is much interest in the "Taelman class module", however this is not exactly the analog of the Selmer group I'm looking for.