Geometrically unirational varieties that are not unirational By a variety over a field $k$, I mean a scheme that is separated and
of finite type over $k$. I indicate changes of the base ring by
subscripts.

Does there exist a smooth and projective variety $V$ over some field $k$, with
$V(k)\neq\emptyset$, that is geometrically unirational, i.e.,
there exists a dominant rational map $$\mathbf{P}^n_{\overline{k}}
\dashrightarrow V_{\overline{k}}$$ for some integer $n$, but not
unirational, i.e., there does not exist a dominant rational map
$$\mathbf{P}^n_k \dashrightarrow V$$ for any integer $n$.

I don't recall ever coming across an example of a $V$ as in the question. It is a classical fact that any such $V$ has to have dimension at least $2$. Furthermore, I think I would have known if any examples existed where the dimension of $V$ is $2$. On the other hand, while the condition $V(k)\neq\emptyset$ is
certainly needed to keep the question from having a trivial answer in
the positive (e.g., let $V$ be a smooth cubic surface in
$\mathbf{P}^3_k$ that does not have a $k$-point), it seems too
unnatural to me to be sufficient for a negative answer to the
question in general. 
NB. I realize that some people (including myself at times) use the
term "(uni)rational" for the concept I'm referring to as "geometrically
(uni)rational". However, when comparing the two properties, I somehow
prefer talking about "geometrically unirational vs. unirational" to
"unirational vs. unirational over the ground field".
Some edits made to reflect comments by Jason Starr and ayanta; see below.
 A: Here is a sketch of a counterexample, but I will need to add more details.  Let $k$ be an infinite field.  Let $G$ be a (smooth, connected) semisimple algebraic group scheme of adjoint type over $k$.  Assume that $G$ is "quasi-split", i.e., there exists a closed smooth subgroup scheme $B$ in $G$ that is smooth over $k$ and whose base change to an algebraic closure of $k$ is a maximal, connected, solvable subgroup scheme.  Let $\mathcal{T}_B$ be a $B$-torsor over $k$ whose associated $G$-torsor, $\mathcal{T}_G$, is nontrivial.  Obviously I need to prove that there exists such a $(G,B,\mathcal{T}_B)$, but let's assume for the moment that it exists.
There is a "wonderful compactification" $\widehat{G}$ defined over $k$ that contains a copy of $G$ as a dense open subscheme and such that the natural action of $G\times G$ on $G$ (by left and right multiplication) extends to all of $\widehat{G}$.  The scheme $\widehat{G}$ is smooth and projective over $k$.  The minimal $G\times G$-orbit in $\widehat{G}$ is isomorphic to $(G/B)\times (G/B)$.  Using the left action on $\mathcal{T}_G$ and the right action on $\widehat{G}$, form an action of $G$ on $\widehat{G}\times \mathcal{T}_G$.  Since the action of $G$ on $\mathcal{T}_G$ is free, so is the action on $\widehat{G}\times \mathcal{T}_G$.  
The  geometric quotient $\widehat{\mathcal{T}_G} = (\widehat{G}\times \mathcal{T}_G)/G$ is a smooth, projective $k$-scheme that contains $\mathcal{T}_G$ as an open subscheme and is geometrically isomorphic to $\widehat{G}$.  The stratum $(G/B)\times (G/B)$ becomes $(G/B)\times (\mathcal{T}_G/B)$.  The point is, since $\mathcal{T}_G$ is induced from the $B$-torsor $\mathcal{T}_B$, the quotient $\mathcal{T}_G/B$ has a $k$-point, namely the image of $\mathcal{T}_B$.  Thus, the "deepest stratum" in $\widehat{\mathcal{T}_G}$ has a $k$-rational point.  Yet, since $\mathcal{T}_G$ is nontrivial, there are no $k$-rational points in this dense open subset.  Over an infinite field $k$, this is enough to insure that $\widehat{\mathcal{T}_G}$ is not unirational.  However, it is geometrically isomorphic to $\widehat{G}$, which is rational.
$\textbf{Edit}$.  The "details" above turn out to be impossible in the adjoint case, as Xuhan points out: there is no such triple $(G,B,\mathcal{T}_B)$.  In the intermediate case, it may be possible.  However, even for perfect fields (where there is no issue with constructing wonderful compactifications via descent), the wonderful compactification of a non-adjoint group tends to be singular (although geometrically normal).  
A: Rosenlicht found forms of $\mathbb{G}_a$ over a nonperfect field $k$ that have only finitely many points, hence are not unirational over $k$ (but, of course, become rational over the algebraic closure of $k$).
