Suppose we have a domain $\Omega\subset \mathbb{R}^n$ which is homeomorrphic to the unit ball $B(0,1)\subset \mathbb{R}^n$ and such that $\partial \Omega$ is of class $C^1$ (technically, this means that for every point in the domain we can give a $C^1$-diffeomorphism to the half-space in dimension $n$).

A map $f\colon (X,d_X)\to (Y,d_Y)$ between metric spaces is said to be bi-Lipschitz if there is a constant $K>0$ such that $$ \frac{1}{K} d_X(x_1,x_2) \leq d_Y(f(x_1), f(x_2)) \leq K d_X(x_1,x_2)$$ for all $x\in X$. The spaces $X$ and $Y$ are said to be Lipschitz equivalent if there is a surjective bi-Lipschitz map between them (any bi-Lipschitz map is necessarily injective).

*I wonder if in this case the domain $\Omega\subset \mathbb{R}^n$ is Lipschitz equivalent to the unit ball $B(0,1)\subset \mathbb{R}^n$. If true, is the condition $C^1$ necessary, sufficient?*

EDIT: My original question did not assume that the domain was homeomorphic to the ball. However, in this case it is clearly false, so I added this condition.