Is the inverse of a smooth bi-Lipschitz map continuously differentiable?

Question

Let $f:\mathbb{R}^d\to\mathbb{R}^d$ be smooth (i.e. $C^\infty$) and bi-Lipschitz. Is the inverse $f^{-1}$ in $C^1$? If we knew that $\nabla f$ was invertible, this would follow immediately from the inverse function theorem. However, in this setting I am not sure if the statement is true and have not been able to prove it. Also, by Rademacher's theorem we get that the inverse is differentiable a.e., but not continuous differentiability (continuous differentiability a.e. would suffice, however).