= Solution
For $|z|<1$ on the negative-real portions of the <Hankel contour>, expand
$$
\frac{z}{e^{-t}-z}=\frac{ze^t}{1-ze^t}
=\sum_{n\geq1}z^ne^{nt}.
$$
Local uniform convergence permits termwise integration. The reciprocal Hankel formula gives
$$
\frac1{2\pi i}\int_{-\infty}^{(0+)}e^{nt}t^{s-1}dt
=\frac{n^{-s}}{\Gamma(1-s)},
$$
so $I(z,s)=\sum_{n\geq1}z^n/n^s=\operatorname{Li}_s(z)$ initially on the disc.
For $z\notin[1,\infty)$, choose the contour around the negative axis so that it passes between zero and every pole satisfying $e^{-t}=z$. On compact subsets of the slit $z$-plane it can be chosen uniformly, and differentiation under the integral proves holomorphy. Deforming without crossing a pole gives a single-valued analytic continuation. Near the cut, the pole $t=-\log z$ lies on opposite sides of the two admissible contours; this is why the contour must pass it consistently.
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