= Solution
For a continuous closed path $\gamma:[0,1]\to\mathbb C\setminus\{0\}$, choose a continuous argument lift
$$
\frac{\gamma(t)}{|\gamma(t)|}=e^{2\pi i\theta(t)}.
$$
Its <winding number of a continuous closed path> about zero is
$$
w(\gamma,0)=\theta(1)-\theta(0)\in\mathbb Z.
$$
For a piecewise smooth path this equals $(2\pi i)^{-1}\int_\gamma dz/z$.
If $|\gamma(t)|>|\phi(t)|$, then
$$
H(s,t)=\gamma(t)+s\phi(t),\qquad 0\leq s\leq1,
$$
never vanishes, since $|H(s,t)|\geq|\gamma(t)|-s|\phi(t)|>0$. Thus $H$ is a homotopy through closed paths avoiding zero. By <homotopy invariance of winding number>, or directly by the <dominated perturbation preserves winding number>[dominated-perturbation lemma],
$$
w(\gamma+\phi,0)=w(\gamma,0).
$$
More generally, $\gamma_0$ and $\gamma_1$ are homotopic by paths in $\Gamma$ when there is a <continuous function>
$$
H:[0,1]^2\longrightarrow\mathbb C\setminus\{0\}
$$
with $H(0,t)=\gamma_0(t)$, $H(1,t)=\gamma_1(t)$, and $H(s,0)=H(s,1)$ for every $s$. The winding-number theorem states that such a homotopy implies
$$
w(\gamma_0,0)=w(\gamma_1,0).
$$
For the <Fundamental theorem of algebra>, let $P(z)=a_nz^n+\cdots+a_0$ with $n\geq1$. For sufficiently large $R$,
$$
|a_nR^ne^{2\pi int}|>
|a_{n-1}R^{n-1}e^{2\pi i(n-1)t}+\cdots+a_0|
$$
for every $t$. The dominated-perturbation lemma shows that $t\mapsto P(Re^{2\pi it})$ has the same winding number as $t\mapsto a_nR^ne^{2\pi int}$, namely $n$. If $P$ had no zero, however,
$$
H(s,t)=P(sRe^{2\pi it})
$$
would be a homotopy in $\Gamma$ from that loop to the constant loop $P(0)$, whose winding number is zero. This contradiction proves that $P$ has a complex root. This is the <winding-number proof of the fundamental theorem of algebra>.
Finally suppose that a continuous retraction $r:D^2\to S^1$ existed. The boundary loop $\eta(t)=e^{2\pi it}$ has winding number one, while $(1-s)\eta(t)$ contracts it to zero inside the <disc>. Composing this contraction with $r$ gives a homotopy through loops in $S^1$ from $r\circ\eta=\eta$ to the constant loop $r(0)$. Their winding numbers are respectively one and zero, contradicting homotopy invariance. Hence there is no such retraction, as in the <winding-number proof of the no-retraction theorem>.
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