0708-1300/the unit sphere in a Hilbert space is contractible

Let $H=L^2[0,1]$ and define $S^{\infty}=\{x\in H| ||x||=1\}$

Claim

$S^{\infty}$ is contractible

Proof

For any $t\in[0,1]$ and any $f\in H$ define $f_t(x)= f$ for $0\leq x \leq t$ and $f_t(x)=1$ for $t<x\leq1$. Observe that $t\rightarrow f_t/||f_t||$ is continuous and gives the desired retraction to the point $f=1$.

This proof only works in separable Hilbert spaces? Is the unit ball in a non-separable Hilbert space contractible?

The answer seems to be YES see Spheres in infinite-dimensional normed spaces are Lipschitz contractible