Which class of reflexive spaces $X$ having the property: if a sequence $(x\_{n})\_{n}\subset B\_{X}$ converges to $x$ weakly and $\|x\_{n}\|\rightarrow 1$, then the norm of $x$ must be 1. Of course, the classical sequence spaces $l\_{p}$ do not have this property.It seems that this condition is too strong.



A question about weak convergence on the unit ball of a reflexive space
I think the following space which is called Grothendieck space may be partial answer for your question. 


<https://en.wikipedia.org/wiki/Grothendieck_space>


