Proof:

Suppose we have a retraction $r : E^2 \longrightarrow S^1$ then the induced map

$$r_{*}:\pi_1(E^2,1) \longrightarrow \pi_1(S^1,1)$$

would be surjective which means we have a surjective group homomorphism

$$r_{\ast}:\{1\} \longrightarrow \mathbb{Z}$$

which is impossible. $\square$