Is there a self-map of the disk with Jacobian everywhere greater than $1$? It might be silly, but I am not sure how to approach this problem.

Let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Does there exist a smooth map $f:D \to D$ such that $\det df >1$ everywhere?

I don't assume $f$ maps boundary to boundary. 
The area formula implies that such a map cannot be injective.
I guess this is somewhat related to the question of existence of maps of non-zero degree between disks. 

Choose an $N\gg1$, and map $D$ to an ellipse $E\subset R:=[-2N,2N]\times\bigl[-{1\over N},{1\over N}\bigr]$ by putting
$$f_1:\quad (x,y)\mapsto \bigl( 2Nx,{1\over N}y\bigr)\ .$$
We have ${\rm det}(df_1)=2$. Now use a map $f_2$ with ${\rm det}(df_2)\approx1$ to wrap the long rectangle  $R$ essentially area preserving   around the annulus
$$A_\rho:=\bigl\{(x,y)\in D\bigm| \rho\leq\sqrt{x^2+y^2}\leq1\bigr\}$$
with $\rho$ slighty smaller than $1-{2\over N}$.
