There is some work which generalises the usual Wilson loop in QFT to higher dimensions and constructs non-abelian Wilson surface functionals in the context of non-abelian gerbes.

It seems to me that the context of this work is mostly geometry/topological in nature, where the aim is ultimately to try and rigorously define four-dimensional invariants of knotted surfaces.

However, I am not really clear on what the exact physical application is of these Wilson surface observables besides knot invariants. Do they play any more concrete or direct physical role in calculations of amplitudes similar to regular Wilson loops in gauge theory, or are they used to study topological QFTs or string theory?

Edit: Besides the works mentioned in the answer, I also found this article of Gukov and Witten which mentions quite a large number of applications of these surfaces.