  In this study, we analyze the various strengthening and weakening of the
uniqueness of the Hahn--Banach extension. In addition, we consider the case in
which $Y$ is an ideal of $X$. In this context, we study the property-$(U)/
(SU)/ (HB)$ and property-$(k-U)$ for a subspace $Y$ of a Banach space $X$. We
obtain various new characterizations of these properties. We discuss various
examples in the classical Banach spaces, where the aforementioned properties
are satisfied and where they fail. It is observed that a hyperplane in $c_0$
has property-$(HB)$ if and only if it is an $M$-summand. Considering $X, Z$ as
Banach spaces and $Y$ as a subspace of $Z$, by identifying
$(X\widehat{\otimes}_\pi Y)^*\cong \mathcal{L}(X,Y^*)$, we observe that an
isometry in $\mathcal{L}(X,Y^*)$ has a unique norm-preserving extension over
$(X\widehat{\otimes}_\pi Z)$ if $Y$ has property-$(SU)$ in $Z$. It is observed
that a finite dimensional subspace $Y$ of $c_0$ has property-$(k-U)$ in $c_0$,
and if $Y$ is an ideal, then $Y^*$ is a $k$-strictly convex subspace of
$\ell_1$ for some natural $k$.
