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.