We have not been given enough information to determine whether the area that is "conversationally" described with a phrase that includes the "K" word really does satisfy the NEC definition of "kitchen." If for example there is no sink then the matter is resolved: it is not a kitchen and therefore does not require AFCI. To use my own home as another example, I have what I like to call an "outdoor kitchen." It is about 20 feet away from the rear patio door, and does not share a wall with the house. In this area I keep and use a gas grill, a charcoal grill, a smoker, and a turkey fryer, not one item of which constitutes a "permanent provision for cooking." So that area is not a "kitchen" by the NEC definition. If the gas grill was of the type that is permanently mounted, instead of having wheels, then the issue could be open to debate.
So let me toss a question back to joecalvin: what sort of cooking equipment do these areas have, and can that equipment be classified as "permanent"?