Let $S$ be a smooth affine variety over $\mathbb C$ and let $f:X\to S$ be a finite unramified morphism.

Suppose that $X(K(S))$ is non-empty. (This means that $X\to S$ has a section generically. It does not imply $X\to S$ being generically trivial.)

Is $X(S)$ non-empty?

The answer is positive if $S$ is of dimension at most one.