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If you look at the definition of (strongly) locally uniformly convex space in Diestel, Geometry of Banach spaces, Section 2 of Chapter II, you will see that this is what is needed for the desired property to hold. Strict convexity is not enough. And there are examples of strictly convex Banach spaces which are not locally uniformly convex, you can even re-norm $\ell_2$ to achieve this.