$R$ is a local commutative integral domain that is not a field with maximal ideal $m$ and fraction field $F$. Is it possible that there is an isomorphism $R/m \cong F$?
Equivalently, is it possible that there is a surjection $R \rightarrow F$ or that there is an injection $R \rightarrow R/m$?
Let $F=\Bbb Q(X_1,X_2,\ldots)$ be the field generated over $\Bbb Q$ by an infinite sequence of transcendental elements. Then $F\cong F(X)$. Now take $R$ to be the localisation of the polynomial ring $F[X]$ at the prime ideal $(X)$. The residue field is $F$ and the field of fractions is $F(X)$.