Examples abound, but I’d say my favourite is the electron g-factor which is an example of one of the most precise agreements between theory and experiment ever achieved by humans and it’s calculation relies heavily on the quantum mechanics of the electromagnetic field.
But there’s lot of other, more pedestrian examples. For example, you can (and people have) generated entangled pairs of photons and then performed measurements on this entangled pairs that violate the Bell inequalities, meaning that their correlation is not classical.
Looking back on this, I feel I should also mention that there's not any single piece of evidence that should be bullet-proof convincing evidence for the EM field's quantum nature, rather it's the totality of many many many different, independent results exploring various quantum mechanical properties of electromagnetism (and its high energy unification with the weak nuclear force), all of which converge on the basically undeniable fact that electromagnetism is a quantum mechanical phenomenon.
Of course, this isn't the full story. There is physics beyond the standard model. But whatever that physics is, it's description needs to reproduce quantum electrodynamics in the appropriate limit.
I should also mention that it's not even mathematically consistent to have a classical field like electromagnetism coupled to quantum matter. It's fine in certain approximation schemes, but the breakdown and limitations of such an approach is well documented.
