There could be a soffit there.

Or the ceiling could slope down toward the outer wall.

Or, actually, the top of the cooktop could be 37" or a bit higher. Many cooktops sit fairly high off the counter now, especially gas. So this could be a 7 foot ceiling.

If it is an area of ceiling height over something built in like cabinets or fixtures, it could be allowable in some areas.

Or it could be a kitchen that has been grandfathered in.

There are many kitchens and baths in my area that have ceilings that are 6'8" to 6'10" --just high enough to accomodate a standard door.

These were built in apartments where lofts or two levels were inserted in old, high ceilinged buildings in the 1960s, and they number in the thousands. I lived in one of them. The kitchen level was 6'8" and there were a series of closets, a hall and a powder room with 6'7" ceilings underneath. There were at least a dozen of those apartments in my complex, and there were several identical blocks of buildings with similar layouts. Similarly, there are thousands of trinity houses with lean-to kitchens and baths with low ceilings. None of this would be allowable as new construction, but it it all grandfathered even with major renovations because to alter it would require the actual demolition of thousands of dwelling units.

So, technically it is an allowable legal ceiling height, under certain conditions.

Anyway, I forgot to answer the question. I would use a standard under cabinet style hood and box out the top rather than using a chimney style.

Sophie Wheeler, there is a soffit. The height from the floor to bottom of soffit is 84".

palimpsest, what do you mean by "box out the top?"