A ruled residue theorem for function fields of hyperelliptic curves

We study residually transcendental extensions of a valuation v on a field E to function fields of hyperelliptic curves over E. We show that v has at most finitely many extensions to the function field of a hyperelliptic curve over E, for which the residue field extension is transcendental but not ruled, assuming that the residue characteristic of v is either zero or greater than the degree of the hyperelliptic curve.