Is it possible to write $\tan^{-1}(x)$ as a power series of $\tanh(x)$?

Let $u=\tanh x \iff \tanh^{-1}u=x$. Then it is enough to expand $\tan^{-1}\tanh^{-1}u$ around $u=0$.

You will find that $\tan^{-1}\tanh^{-1}u=u+\frac{u^5}{15}+\frac{u^7}{45}+\frac{64u^9}{2835}+O(u^{11})$, and thus

$$\tan^{-1}x=\tanh x+\frac{(\tanh x)^5}{15}+\frac{(\tanh x)^7}{45}+...$$