The question is basically in the title.

My naive thought was that when a commutation relation holds for all field operators $\Psi(\vec{x})$ (by "all" I mean "at all positions $\vec{x}$") on a fixed time-slice, then it should automatically also apply to the spatial derivatives of those field operators, $\partial_{\vec{x}} \Psi$.

My reasoning to that is that a derivative is the limit of the difference of two operators at two proximate positions. Since for the spatial derivatives, all operators in question lie in the same time-slice, they all commute / anticommute, and hence their difference should do so as well.

Is that true, or are there some issues with infinities that I didn't take into account here?