Question: Is the image of a contractible space under a contraction contractible?

More specific to what I'm looking for: is the image of the unit ball $B\_1(0)\subset\mathbb{R}^n$ under a contraction mapping $F:\mathbb{R}^n\to\mathbb{R}^n$ still contractible?

Answer: 
Consider the smooth surjective exponential map, which is Lipschitz continuous $e: T\_S S^n = \mathbb{R}^n \rightarrow S^n$ ($S$ being the south pole). 

Consider a large $M>0$ and a small $r > 0$ and define $F(x\_1, \ldots, x\_{n+1})=r \cdot e(Mx\_1, \ldots, Mx\_n) \in S^n \subset \mathbb{R}^{n+1}$. 

If $M$ is large enough, then $F(B\_1(0))=rS^n$ for any $r$. If you choose then $r$ small enough, $F$ is a contraction, and the image of $B\_1(0)$ is homeomorphic to $S^n$, which is not contractible.
The map $H\_t(x)=tF(x), t\in [0,1]$ restricted to the unit ball defines a deformation retract of the unit ball to a point.