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Am on my phone, so just very briefly: yes.
In fact the running assumption in textbooks that physical fields are plain functions is generically violated unless one stays either with toy systems or in perturbation theory. For instance gauge fields are, nonperturbatively, not functions. One way to model them is as smooth functors on a smooth path groupoid. Hence that is what enters the EL-equations.
Tomorrow when I am properly online I’ll provide citations.
So the formulation of gauge fields as functors that I was thinking of is discussed here. See also the list of related references.
A general way to speak of Euler-Lagrange equations – hence of critical loci – in contexts where physical fields are generalized this way from plain functions is here, though if you want to really work with this you will have to flesh this out more. For the moment this is just to answer your bare question: yes, there are EL equations on spaces of smooth functors, hence whose solutions are smooth functors. In fact the true EL equations for gauge fields are really always of this form, only that traditionally they are truncated to a perturbative sector where the gauge field is regarded as a differential-form valued function.
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