# Is the sheaf of non-vanishing holomorphic functions soft?

It is obvious that the sheaf of holomorphic functions is not soft, not every holomorphic function on a closed set (appropriately defined) can be extended to be an entire function. What about the sheaf of non-vanishing holomorphic functions $$\mathcal{O}_X^{\ast}$$? I assume that it is not soft either, but I'd like an example.

The case of non-vanishing holomorphic functions is really only easier. For instance, consider the function $$f(z)=z$$, defined on a non-discrete closed set that does not contain $$0$$. Since a holomorphic function on a connected set is determined by its values on any non-discrete set, the only possible extension to all of $$\mathbb{C}$$ is $$f(z)=z$$, but this is not a non-vanishing entire function.