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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.

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    Dear Haki, Do you know any examples of singular rational curves, e.g. as alluded to in QiL's answer below? Regards,2011-12-16

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If $X$ is any model of ${\bf P}^1_K$ over $C$, then any morphism ${\bf P}^1_C\to X$ is birational, quasi-finite (the fibers are finite sets) hence finite because ${\bf P}^1_C$ is projective over $C$. So if $X$ is normal, then ${\bf P}^1_C\to X$ is an isomorphism.

If you don't require $X$ be normal, there are counterexamples. Let $Y$ be any singular integral rational curve, let $X=Y\times_k C$ (here $k$ is the field on which $C$ is defined). Then $X$ is a non-normal (hence non-smooth) model of ${\bf P}^1_K$. The normalization map ${\bf P}^1_k\to Y$ induces a surjective birational morphism ${\bf P}^1_C\to X$.