Probably this paper answers your question negatively.
In particular it shows that if $X$ is an open subset of $\mathbb{C}^n$ and the sheaf $\mathcal{O}$ of holomorphic functions on $X$ is given the topology of compact convergence then every continous endomorphism of $\mathcal{O}$ is given by a convergent $\mathcal{O}$-linear sum of operators $\partial_1^{\alpha_{1}}\cdots \partial_n^{\alpha_n}$ with $\alpha_i\in \mathbb{N}$.