Let $K$ be a number field and $S$ a finite set of places of $K$. Then Shafarevich's theorem states that there are only finitely many isomorphism classes of elliptic curves $E$ over $K$ with good reduction outside $S$.

My question is whether the above holds when $K$ is the function field of an algebraic curve over an algebraically closed field of characteristic zero, and $E / K$ has non-constant $j$-invariant?