Let $k$ be a perfect field.

We say that $k$ has property $(*)$ if for every geometrically irreducible morphism locally of finite type $X\rightarrow \mathrm{Spec}\, k$ admitting a section, the set of points $x\in |X|$ such that the canonical injection $k\rightarrow k(x)$ is a surjection is dense.

Do there exist non-algebraically closed fields satisfying $(*)$?