  We prove that there is a unique $R$-equivalence class on every del Pezzo
surface of degree $4$ defined over the Laurent field $K=k((t))$ in one variable
over an algebraically closed field $k$ of characteristic not equal to $2$ or
$5$. We also prove that given a smooth cubic surface defined over
$\mathbb{C}((t))$, if the induced morphism to the GIT compactification of
smooth cubic surfaces lies in the stable locus (possibly after a base change),
then there is a unique $R$-equivalence class.
