A lipschitz function $f(x)$ valued in the unit ball has almost everywhere Frechet derivative orthogonal to $f(x)$ if $\|x\|=1$.

Question

Consider a function $f:R^n\to R^m$ valued in the unit ball $B=\{u\in R^m: \|u\|=1\}$.

Assume $f$ is Lipschitz. By Rademacher's theorem, $f$ is differentiable almost everywhere, i.e., for almost every $x$ there exists a Jacobian matrix $D_f(x)$ such that $$ f(x+h) = f(x) + D_f(x) h + o(\|h\|). $$

Is it always true that for almost every $x$, $\|f(x)\|=1$ implies $f(x)^T D_f(x) = 0$?

Proof: (is it correct?): if $\|f(x)\|=1$ and $f(x)D_f(x) \ne 0$ then $\|f(x+h)\|^2 = \|f(x)\|^2 + 2 f(x)^T D_f(x) h + o(\|h\|)$ and we can escape the unit ball in the direction $h = D_f(x)^T f(x)$, a contradiction. Hence $f(x)D_f(x) =0$ necessarily.