Let $C$ be a smooth curve with function field $K$. The projective line $\mathbf{P}^1_C$ is a smooth model over $C$ for the projective line $\mathbf{P}^1_K$. Does there exist another model of $\mathbf{P}^1_K$, say $X$, and a surjective birational morphism $\mathbf{P}^1_{C} \to X$?
On the generic fibre this morphism is the identity.