OK, try this. When an electron and a positron annihilate, their fields annihilate, too. We go from some field--with a vanishing monopole moment in the far field, but with a substantial dipole moment, right up to the end--to no field at all. (I hope you will agree that once the particles are annihilated, their field suddenly disappears.) The collapse of the field demonstrably travels outwards at the speed of light, because the collapse of the field physically manifests itself in the form of photons, as is demanded by quantum mechanics (no relativity involved, here).
But how do we know that those photons travel at the speed of light? You haven't actually measured their velocity, have you? Actually, we have. One useful technique in astronomy is to look at the annihilation photons coming from distant galaxies and nebulae. It tells us where the really hot regions are: antimatter is not easy to come by! If the information of the field destruction came to us at any velocity other than c, the annihilation photons wouldn't line up with the source in visible light. This is not the case.
Man, don't tell Boris!
OK, so the field collapses, and we see a non-field entity called a proton emit outwards from this collapse at the speed of Light.
Now, can we demonstrably show that the Field itself is collapsing at the speed of Light, rather than that the *effects* of the Field collapsing emit themselves at the speed of Light?