Let $S$ be an integral 1-dimensional scheme with function field $K$.
Let $E$ be an elliptic curve over $K$. The torsion of $E$ over $K$ is not necessarily finite. As an example consider an elliptic curve over $\mathbf C(t)$.
Now, assume the residue field of each closed point of $S$ to be finite.
Is the torsion of $E(K)$ finite?
Note that I'm not assuming $S$ to be noetherian.
Note that we may assume $S$ to be affine. To answer the above question positively, it would suffice to show the torsion embeds into the rational points of a special (not geometric) fibre of $E$.