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: 
No. For instance, the map 
\begin{align\*}
F: \mathbb{R}^2 &\to \mathbb{R}^2 \\
(x,y) &\mapsto \frac{1}{4\pi}\left(\cos(2\pi x), \sin(2\pi x)\right)
\end{align\*}
takes the unit ball to a circle, which is not contractible. That the map is a contraction can be seen by the derivative, or by noticing that it is the composition of the projection in the first coordinate with a walk on the circle with constant velocity equal to $1/2$. (This should make for a quick visualization that it is indeed a contraction.)
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.