Analytic rank-one elliptic curves over function fields and their rank over certain ring class fields

Let $E\mathrm{/}k$ be a non-isotrivial elliptic curve over a global function field $k$ of characteristic $p>3$, and $G\subset \mathrm{G}\mathrm{a}\mathrm{l}(k^{\mathrm{s}\mathrm{e}\mathrm{p}}\mathrm{/}k)$ be a topologically finitely generated subgroup. We prove that if $E\mathrm{/}k$ has analytic rank $1$, then its rank over the fixed subfield $L^G$ is infinite, where $L$ is the infinite ring class extension of some finite separable extension $K\mathrm{/}k$. If $E\mathrm{/}k$ has analytic rank $0$, then we prove that the same holds provided there exists an imaginary quadratic extension $K\mathrm{/}k$ such that $E\mathrm{/}K$ has analytic rank $1$ and satisfies the Heegner hypothesis.