Hahn-Banach extensions from $E$ to $E^{**}$.

$C(K)$ for $K$ compact, Hausdorff, infinite satisfies the requirements. From arXiv:math/9605213 Remark 7, we see

For $C(K)$ spaces, the property of being nicely smooth is equivalent to reflexivity.

Further, it is well known that for $C(K)$ spaces, $K$ finite is equivalent to reflexivity.

Finally, Remark 4 in the same paper states that

Hahn-Banach smooth spaces are nicely smooth

where Hahn-Banach smooth spaces are exactly those for which every functional on $X$ Hahn-Banach extends uniquely to $X^{**}$. The paper however does not justify this result, though this paper attributes it to Godefroy:

Godefroy, G., Nicely smooth Banach spaces, in “Texas Functional Analysis Seminar 1984–1985”, (Austin, Tex.), 117 – 124, Longhorn Notes, Univ. Texas Press, Austin, TX, 1985

I can't find an online version of this, nor can I prove myself that Hahn-Banach smooth spaces are nicely smooth (admittedly, I haven't tried much). Many other papers seem to cite this result however, so it seems reasonably reliable.

It took me some time to come back to this question but I've found an answer that somewhat satisfies me.

Proposition: Let $E$ be a Banach space and denote as $J_E:E \to E^{**}$ its inclusion in its double dual. Then given a norm one functional $\varphi \in E^*$ the following are equivalent.

• $J_{E^*}(\varphi)$ is the unique functional of norm one in $E^{***}$ that extends $\varphi$ from $J_E(E)$ to $E^{**}$.

• The identity map $(B_{E^*},w^*) \to (B_{E^*},w)$ of the unit ball of $E^*$ with the weak-$*$ topology to itself with the weak topology is continuous at $\varphi$ .

An immediate corollary is

Corollary: Let $E$ be a Banach space. The following are equivalent.

• Every functional in $E$ has a unique norm-preserving extension to $E^{**}$.
• The $w$ and $w^*$ topologies coincide in the unit ball of $E^*$.