Can a non-proper variety contain a proper curve

Question

Let $f:X\to S$ be a finite type, separated but non-proper morphism of schemes.

Can there be a projective curve $g:C\to S$ and a closed immersion $C\to X$ over $S$?

Just to be clear: A projective curve is a smooth projective morphism $X\to S$ such that the geometric fibres are geometrically connected and of dimension 1.

In simple layman's terms: Can a non-projective variety contain a projective curve?

Feel free to replace "projective" by "proper". It probably won't change much.

Answer 1

Sure, $\mathbb{P}_k^2-\{pt\}$ is not proper over $Spec(k)$ and contains a proper $\mathbb{P}_k^1$.