You can get very slim extractors that might do the job:
https://pando.es/en/hoods/e-240/

Or extractors that are designed to be seen:
https://pando.es/en/hoods/i-898/

https://luxairhoods.com/Ceiling-Cooker-Hoods/ceiling-cooker-hoods-slimline

Would these work?
Presumably it is the motor unit that you can’t sink into the ceiling.
Can’t you build your box deeper than 15cm to house the motor unit and then duct in the joist space?
If not and you need a recirculating model look at the Elica Cloud 7.

Do you want an extractor at all, the pendant lamp models are only a recirculating extractor and expensive so you'd be better off with a 60 litre/sec vent-axia type on an external wall. Even with an island chimney extractor you wouldn't be able to duct it out by the sounds of it unless you're in a bungalow.

Not sure if relevant depending where you are and if part of much bigger work. We added an extension and needed uilder control sign-off at the end.  They wouldn't allow for recirculation hood and needed it to be an external venting hood.  This could be because it was an new extension/room and not remodelling an existing room.  Or just local council.  If this is part of a bigger extension and you need building control sign-off at the end.. check they allow recirculating.  (It was a condensation question for ours from steam.. not about kitchen cooking smells)

Regarding building regs…… they require kitchens and bathrooms to be vented so you are supposed to have an extractor, but that doesn’t have to be over the hob. So satisfy building regs with an inexpensive extractor elsewhere and if whatever appliance you choose to put over the hob is unvented at least you have a grease filter near your cooking area.

Hi all
This is not an extension. We just converted the living room into the kitchen

As a rule of thumb, you shouldn't make the extraction any worse than what was existing when refurbishing a kitchen, but part F building regs states any new kitchen should be provided with a mechanical extract fan, so is moving the kitchen to a new part of the house considered as being a new installation or a refurbishment.

As I understand it (but check with local building control), providing a kitchen or bathroom in a room where there wasn't one before, building regs approval is likely to be required to ensure the room will have adequate ventilation, drainage and meets the requirements in respect of structural stability, electrical and fire safety.

You would need either a 30 litre/second extractor if its over the hob or 60 litre/second if its placed elewhere ie on an external wall to meet part F. But you may also have to consider/meet some of the other parts of the regs. as well.

Be interesting to know if you do ask building control what their answer is (get it in writing from them so they can't change their mind).

We are having the same work done as you … only at the start though… we are getting a recirculating neff extractor that will sit in a suspended box above the island as to avoid joists etc the distance to vent was too far. I know this is what is agreed - but not sure on the depth the box will drop down. 50cm sounds a lot

Hi, you can box down the same shape as the island and put a ceiling hood in the boxing? duct inside the boxing and into the ceiling space and out?

We work with various extractor companies including Elica, Neff, Bora and Novy. Give us a call is probably your best bet, 01535 282021, and we can discuss the options or if you let me know your email address, then I can send you some thoughts. :)

Lifestyle Kitchens sound like they're talking sense. Personally I wouldn't bother with an extractor there and use the space more creatively, then stick one in on a wall somewhere like in these photos. But each to their own.