= Solution
The disc lies in the upper-right half-plane and avoids zero and the negative real axis. On the slit plane take the <holomorphic logarithm> $\operatorname{Log}z=\log|z|+i\arg z$ with $-\pi<\arg z<\pi$. Its derivative is $1/z$, and $e^{\operatorname{Log}z}=z$. Therefore the <holomorphic square root>
$$
\boxed{f(z)=\exp\left(\frac12\operatorname{Log}z\right),\qquad z\in D,}
$$
is analytic and satisfies $f^2=z$. One can also construct it without a global branch theorem: fix either scalar square root $a$ of $3+3i$ and expand $a(1+(z-3-3i)/(3+3i))^{1/2}$ by its convergent binomial series. The series converges on the entire disc because $|z-3-3i|/|3+3i|<1/\sqrt{18}$. Either sign gives an admissible choice.
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