I think this answer is morally the same as @fedja's above, but maybe slightly more explicit. Set $f(x)=x^4\sin(1/x)/(1+x^4)$, so that it looks like $x^4\sin(\pi/x)$ near the origin and define $h(x,y)=(x,y+\epsilon f(x))$. This is a global $C^2$ diffeomorphism and satisfies $\|D_xh(v)\|\ge c_\epsilon\|v\|$ for all $x$ and $v$ for a uniform $c_\epsilon$. Choose $\epsilon$ such that $c_\epsilon>1/\sqrt 2$. Now set $T(x,y)=(2x,32y)$ and consider $\tilde T=h^{-1}\circ T\circ h$. Now iterates of the line under $\tilde T$ are conjugate by $h$ to iterates of the graph of $f$ under $T$, so it suffices to show that iterates of the graph of $f$ under $T$ become dense in the unit ball.
In the range $x\in (2^{-(n+1)},2^{-n}]$, the graph has $2^n$ zeros. The height of the graph is approximately $\pm 2^{-4n}$ between each pair of zeros. Iterating $n$ times, the graph reaches height $2^{5n}\times \pm 2^{-4n}$, so that the $n$th iterate of the graph crosses $2^n$ times between roughly $\pm 2^n$ for $x$ values in the range $[1/2,1]$. In particular, the iterates become dense in the unit ball.