Proof:

Assume $n\geq 2$. A retraction $r:E^n\longrightarrow S^{n-1}$ would imply that $H_{n-1}(r):E^n\longrightarrow S^{n-1}$ is surjective which is plainly false since $H_{n-1}(E^n) = \{0\}$ whereas $H_{n-1}(S^{n-1}) = \mathbb{Z}$. The case $n = 1$ is left to the reader.