Is the $n^\text{th}$ derivative of some function, where $n$ is a matrix, ever defined?

Question

For the function $f(x) = x^a$, the $n^\text{th}$ derivative $\frac{d^n}{dx^n}x^a = \frac{a!}{(a-n)!}x^{a-n}$ . This can be extended to non integer values of $n$ thanks to the gamma function. I recently red that factorials of matrices are a thing, using the gamma function. Also, one can raise a scalar to a matrix power if it is treated as a Maclaurin series. So, if in the above equation $n,a\in\mathbb{R}^{p\times q}$, can we treat it as mathematically correct? More generally, can fractional calculus be extended so that we have matrix-th derivatives or even tensor-th derivatives?