If you have a standard 8' ceiling you can have a ceiling fan -- perhaps even if your ceiling is 7 1/2 feet and you choose a fan that hugs the ceiling; however lower than 7 1/2 that might not be a good idea.

Do measure the actual height of your ceiling. It may be that you need a different kind of fan to circulate the air.

If this one came in white ...

http://homebut.com/blog/fincube-a-brilliant-modular-home/

IXNAY the ceiling fan in a kitchen. All that happens is to blow dust and grease....everywhere.

Sounds like you do a lot of range top cooking with gas. If it comes time for a new range look at the Bosch 800 or Benchmark induction models for a 30" space. Or Bosch and Miele for 36" cooktops. No more waste heat and a cooler kitchen.

Kitchen ventilation isn’t a ceiling fan. It’s an actual kitchen hood with a duct to the exterior that removes the heat, odors, grease, steam, and smoke that cooking produces.

Ventilation is a must to get rid of heat and grease and smells in a kitchen moving all that around does not do anything for heat. Get a proper hood vented to the outside it will make a huge difference. I would never put a ceiling fan in a kitchen

For less money than a ceiling fan, you can purchase an over the range hood and it will work for you provided it is vented to the outside of the home.

While my range hood removes cooking odors, grease, and heat from the range, it does NOT cool off the kitchen and I regret getting rid of the kitchen ceiling fan. What you are looking for is called a "flush-mount" or hugger ceiling fan. Flush-mount ceiling fans

Air conditioning...............................

Agree with oasisowner. You can have all the range venting in the world and that doesn't help if it's hot and humid in the kitchen. And that's even with A/C. You'd be surprised how many kitchens don't have vents where you'd want them, Jan.

But I would want to know what "extremely low" means. If it's, say, 7 feet, I think you're out of luck for a ceiling fan.

Couldn't get that second link pic back either. The ceiling hugging ceiling fan with light was similar to this:

https://www.pinterest.com/pin/154811305919414570/

https://www.pinterest.com/pin/326159197989750479/