The smallest singular value $s_n$, which has been stated to have multiplicity one, is a simple eigenvalue of the matrix $M(z) = P(z)P(z)^*$, or equivalently, a simple root to the characteristic polynomial $\chi(\lambda;z) = \det (\lambda I - M(z))$.

As defined, each coefficient in the matrix $M(z)$ is polynomial with respect to $z$, so $\chi(\lambda;z)$ is a polynomial in $\lambda$ whose coefficients are polynomials of $z$.
$$\chi(\lambda;z) = c_0(z) + c_1(z)\lambda + \cdots + c_n(z)\lambda^n$$

It is a basic result from analysis that the simple roots of a polynomial are smooth functions with respect to the coefficients of the polynomial. And in this case, the coefficients of $\chi(\lambda;z)$ are themselves polynomials of $z$, and hence also smooth. Therefore, the simple roots of $\chi(\lambda;z)$ are smooth with respect to $z$. QED.