Let $f:X\to S$ be a finite type affine morphism of schemes where $S$ is an integral noetherian affine regular scheme whose function field is of characteristic zero.

Assume that all geometric fibers of $f$ are non-empty and that the generic fibre of $f$ is smooth.

Does have $f$ have a section up to replacing $S$ by some fppf covering?

The answer is positive if $f$ is flat, because then $f$ has a section after base-change along $f$.