Has $\frac{z}{\bar{z}}$ complex derivative at $z=0$? Has $\frac{z}{\bar{z}}$ complex derivative at $z=0$? I got zero from contour integral when contour is $z=\gamma(t)= e^{it}$, $t=[0,2\pi]$ so it should be analytic on unit circle.

Have you noticed that
$$
\lim_{\substack{z \to 0 \\ z \in \mathbb{R}}} \frac{z}{\overline{z}} =1
$$
and
$$
\lim_{\substack{z \to 0 \\ z \in i\mathbb{R}}} \frac{z}{\overline{z}} =-1 ?
$$
The fact that the integral along a single, particular loop is zero has no relevance. There is Morera's theorem, but it deals with a continuous function defined in an open domain, and by assumption the integral along any loop must be zero.
