reference request - Space of derivations of holomorphic (analytic) functions

Let $M$ be a (real) smooth manifold, and $p in M$.

The space of (linear) derivations $D:C^{infty}(p) to mathbb{R}$ (i.e., maps satisfying $D(f+g) = D(f)+ D(g)$ and $D(fg)=D(f)g(p) + f(p)D(g)$) on the algebra $C^{infty}(p)$ of differentiable functions defined on some neighbourhood of $p$) is then a $n$-dimensional vector space (this is one way to define the tangent space $T_p M$ after all).

It is easy to see that if we consider instead derivations $D:C(p) to mathbb{R}$ on the space $C(p)$ of continuous functions, then the space of derivations is trivial.

My question is: when $M$ is a complex (or analytic) manifold, what is the dimension of the space of derivations on holomorphic (or analytic) functions defined near p?

I've once heard that this space is infinite dimensional. Is this true? (and if it is, is there a simple proof or some reference material?)