Is every power series infinitely differentiable everywhere? I have found various sources on the internet that say that power series are infinitely differentiable on their interval of convergence:
Wikipedia:

Once a function $f(x)$ is given as a power series as above, it is differentiable on the interior of the domain of convergence.

Northwestern University:

[...] power series are (infinitely) differentiable on their intervals of convergence [...]


But isn't every power series (infinitely) differentiable everywhere?
After all, a power series is just an infinite polynomial and a polynomial of degree $n$ is differentiable $n+1$ times. Source
Doesn't this imply, that a polynomial of "degree $\infty$" is differentiable $\infty$ times?

There is lot of difference between polynomials and power series. 
You  cannot define $\sum z^{n}$ for $|z| >1$. So there is no question of differentiability of this on $\{z: |z| >1\}$. 
