Suppose that $k$ is a perfect field of characteristic $p>0$, $\mathcal{V}$ is a complete discrete valuation ring with residue field $k$ and quotient field $K$, of characteristic $0$.
Then when one does rigid cohomology, or looks at arithmetic $\mathcal{D}$-modules, a hypothesis that often comes up is that of a $k$-variety $X$ being `realisable' or 'embedable', i.e. there exisisting a smooth formal $\mathcal{V}$-scheme $\mathfrak{X}$, topologically of finite type, and a locally closed immersion $X\hookrightarrow \mathfrak{X}$ such that the closure $\overline{X}$ of $X$ in $\mathfrak{X}$ is proper over $k$ (if you want, you could strengthen this to there existing a smooth and proper $\mathcal{V}$-scheme $\mathfrak{X}$ and a locally closed embedding $X\hookrightarrow \mathfrak{X}$).
The notion of being embeddable comes up if you want to define the rigid cohomology of your variety $X$ directly, in general you use the fact that every variety is locally embeddable (since affine varieties are embeddable) to define rigid cohomology by gluing.
My question is: does anyone know of an example of a non-embeddable variety over $k$ (for either the stronger or slightly weaker notion of embeddability)?
There are certainly examples of non-quasi projective varieties, and non-liftable smooth and projective varieties, but obviously being non-embeddable is much stronger than this.
Is anything known if we are allowed to make isometric extensions of $K$?
