A class $\Lambda$ of continua is said to be $C$\textit{-determined} provided that if $X,Y\in\Lambda$ and $C(X)\approx C(Y)$, then $X\approx Y$. A continuum $X$ has \textit{unique hyperspace} provided that if $Y$ is a continuum and $C(X)\approx C(Y)$, then $X\approx Y$. In the realm of metric continua the following classes of continua are known to have unique hyperspace: hereditarily indecomposable continua, smooth fans (in the class of fans) and indecomposable continua whose proper and non-degenerate subcontinua are arcs. We prove that these classes have unique hyperspace in the realm of rim-metrizable non-metric continua.