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 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".

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.

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 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".

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 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".