Let $K$ be the function field of an elliptic curve $C$ over a finite field $\mathbb F_q$, and $|C|$ be the set of closed points of $C$, i.e. the set of places of $K$.
Let $A$ be any element of infinite order in the multiplicative group $(K^\times,\times)$, and hence $A=\dfrac{a}{b}$ for some $a,b\in\mathcal O_K$. So $A\in \mathfrak p$ for any prime ideal $\mathfrak p$ (corresponding to some closed point $p$) not dividing $(b)$. Therefore we can take the reduction of $A$ modulo $\mathfrak p$.
Here $$\mathcal O_K=\{x\in K||x|_\mathfrak p\leq 1\text{ for all }\mathfrak p\}.$$
My question is
Is the following set $$ S=\{n\in\mathbb Z>0 \text{ s.t. there exists some }p\in |C| \text{ with the order of }A\text{ mod }\mathfrak p \text{ in } \mathcal O_K/\mathfrak p \text { is } n \}$$ infinite?
In case $C$ is a projective line, i.e. $K=\mathbb F_q(t)$, It is not hard to show that the set $S$ is infinity by some elementary arguments.
In case $C$ is an elliptic curve, I do not know how to deal with this (and I do not know whether $S$ is finite or not).
This question is a global function field version of Schinzel's theorem for numberfields.