I think 24" hoods are the smallest you will find. Gaggenau used to have a pull up downdraft for the combiset cooktop. That may be an option if you have the countertop space. It appears to be discontinued but you may be able to find them on craigslist or ebay.
Good luck!

Here is a link that might be useful: Gaggenau Pull up Downdraft

Since you want 6" of extension beyond the cooking space to adequately capture the cooking vapors, a 12" cooktop needs a 24" vent.

You could use Broan or other "power packs" or "inserts" which are the working parts of the hood without the hood itself. They are narrower and more discreet. You need to have a custom hood or at least a liner of some sort.

I'm going to have to make my own I guess. I have a similar situation I have a very small kitchen which I cannot expand right now I am putting in a two-burner induction cooktop and I want a very narrow say 12-in pipe that will come down with a possible braun type exhausted insert and hopefully some led light too. Why can't companies listen? With all the tiny homes that are out there people are probably demanding this product and no one is listening 24 inches is just not feasible in my space 18 inches would be tops. Thank thank you for the link to the discontinued item at least that gives me a lead on another option I might have but I'm going to have to make my decision and I've got a vertical cooktop going in so would have to be on the side in a cutout in the countertop. Least situations has to be done in advance so that you can.

You cannot think out of the 24 inch box. Better no hood than a 12 inch oneoneoneoneone