Let $R$ be a DVR of equal characteristic with a perfect residue field $k$. Does there exist a unital homomorphism $k\rightarrow R$ such that the composition with the projection $R\rightarrow k$ is the identity?
If the residue field is imperfect of characteristic $p$ consider $k[x]_{(x^p-a)}$ where $a$ is an element that is not $p$-th power. The residue field is $k(a^{1/p})$ but since the fraction field of $R$ is $k(x)$ it does not contain a $p$-th root of $a$.
If $R$ is of mixed characteristic, then there can not be a unital homomorphism from the residue field to $R$ (never mind the composition with the projection).
What have I tried: if $R$ is complete then it is isomorphic to $k[[x]]$ so the statement is true.
No, consider the DVR $\mathbb{R}[x]$ localized at the the maximal ideal $(x^2 +1)$. The residue field is $\mathbb{C}$ but no subset of $R[x]_{(x^2+1)}$ contains the square root of $-1$.