I will assume $p$ is odd to avoid having to think about places at infinity.

Cassels proved more: There is an exact sequence
$$H^1\bigl(F_\Sigma/F,E[p^\infty]\bigr)\longrightarrow \bigoplus_{v\in \Sigma} H^1\bigl(F_v,E[p^\infty]\bigr)/im(\kappa_v) \longrightarrow S$$
where $S$ is the dual of the projective limit of the $p^n$-Selmer group. The map to $S$ is dual to the limit of the maps
$$ \operatorname{Sel}_{p^k}(E/F) \longrightarrow \bigoplus_{v\in \Sigma} E(F_v)/p^k E(F_v) $$
(The local term here is dual to the one above by local Tate duality). 
The projective limit of the Selmer groups is a finitely generated $\mathbb{Z}_p$-module of rank equal to $\operatorname{rank}(E/F)+\operatorname{corank}(Ш(E/F))$ and the finite torsion subgroup is equal to the $p$-primary torsion of $E(F)$. If you assume that $Ш(E/F)$ is finite, the question becomes what is the kernel of the map from the $p$-adic completion $E(F)\otimes \mathbb{Z}_p$ to the sum of the $p$-adic completions of $E(F_v)$. For $v\nmid p$, these are finite groups, but not for $v\mid p$. A conjecture of Waldschmidt says that the image from the Selmer group to the local terms is as surjective as it is allowed to be, analogous to Leopoldt's conjecture for the $p$-adic regulator of units.

If you climb up a $\mathbb{Z}_p$-tower $F_{\infty}/F$, the chances that $\gamma$ becomes surjective increase because the cokernel will be dual to the the projective limit $\varprojlim_n \varprojlim_k \operatorname{Sel}_{p^k}(E/F_n)$ and, if for instance the rank stabilises in the tower or even better the usual Selmer group is $\Lambda$-torsion, then this projective limit tends to be zero.

All of this is a bit beyond Greenberg's introduction. The proof of Cassels theorem above can be deduced from the global duality results in "Cohomology of number fields".