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Yes. I was under impression a Frechet smooth point would also be an SSD point. Check Proposition 3.1 from the paper by Indumathi & Lalithambigai. For those points $x$, when $d(x,Y)=d(x,B_Y)$ one can have some cases when $P_{B_Y}(x)$ is nonempty or empty.
Slight modification is needed in my last statement. For a point $x\in \ell_1$, $\inf\{\|y\|:y\in P_Y(x)\}\leq 1$ if and only if $d(x,Y)=d(x,B_Y)$. Here $Y$ is as stated above.
My earlier question can be reformulated as follows. Does there exist an equivalent renorming on $C[0,1]$ which makes it (weakly) Hahn-Banach smooth? Probably the answer to this problem is 'No' because the dual of any weakly Hahn-Banach smooth space has RNP.
