  A subspace $X$ of a Banach space $Y$ has $\textit{Property U}$ whenever every
continuous linear functional on $X$ has a unique norm-preserving (i.e.,
Hahn$-$Banach) extension to $Y$ (Phelps, 1960). Throughout this document we
introduce and develop a systematic study of the existence of
$\textit{U-embeddings}$ between Banach spaces $X$ and $Y$, that is, isometric
embeddings of $X$ into $Y$ whose ranges have property U. In particular, we are
interested in the case that $Y=C(K)$, where $K$ is a compact Hausdorff
topological space. We provide results for general Banach spaces and for some
specific set-ups, such as $X$ being a finite-dimensional space or a
$C(K)$-space.
