No. : ) ......and WHY?

At the very least, you need a scaled drawing with every dimension, every inch of where and how you plan to do this. That means every wall, every window, the island ALL of the space,

The wall should hide the stools if this is the way you are going.

Question WHY?

I am more of a fan of having walls instead of open concept but in your case I think the open concept works pretty well. You have generous space between your kitchen island and your sofa. Not sure closing you kitchen off will enhance your space much.

I need the to scale floor plan but from the pics I see no benefit to the addition of a wall only to make the space look smaller

Can you explain the problems you’re trying to solve? noise, visual clutter etc - there may be other solutions (i don’t think walls are it)

You're envisioning something like this? Will there be actual doors or just openings? If doors (unless pocket doors), you'll need to consider where they will swing.

What is the space to the left of this photo? I ask because, assuming the new wall follows the ceiling beam, it will end in thin air on the left. The beam doesn't end where the far wall of the left-side room is. A short wing wall will be necessary, slightly separating the kitchen from the room to the left. (In other words, the new wall will be L-shaped in plan.) Without a floor plan, I can't tell if that will work. Here's a diagram in case this isn't clear:

The other option is to stop the new wall at the island (more or less) and let the remaining opening run all the way up to the ceiling, like this:

To my eyes, that looks odd because of both the ceiling beam and the offset walls. If the ceiling beam is only decorative, not structural, you could perhaps remove it and leave a continuous ceiling over the left-side opening, but the walls will still be offset. Plus you have crown running around the kitchen wall/ceiling junction, but not in the living/dining room. You would probably have to add matching crown to the living/dining.

Some issues to consider, anyway . . .

And to answer the original question, I'd extend the wall past the island to include the bar stool space.

Just wanted to add that we did this as well - put up a wall with two doorways to create a closed kitchen instead of open floor plan. We added great quality custom pocket doors. I'm so glad to have them. They trap kitchen smells, make the kitchen more private when I want to be left alone in there or want to hide a mess from the rest of the dinner party guests, and are great for keeping our dog out of the kitchen when food is cooling on the counter. She's a thief!

If a new wall is going to be a squeeze for the walkways in the kitchen, then don’t do it. Posting a measured layout of the space will get you more informed opinions.

Looking more closely at the original photos, I see that the ceiling heights are not the same on the two sides of the beam, so a continuous flat ceiling over the left-side opening with the beam removed is not possible. It looks like the only option is an L-shaped wall following the beam, then turning to meet the existing walls on the left:

(IMO, that's the best-looking option anyway.)

I see from a previous thread your concern is viewing all the dirty dishes. And if you've just moved in, it's only been a couple weeks if that. Maybe live there a handful of months to see how you function and move about the house. It may not be a concern after all, or at least one that is not more significant than the success of the open concept design in this particular house.

@simplynatural did a very thorough and prescient reply from the info presented. I will add to that if your intent is to leave the island as is, that may not happen. The ogee counter needs replaced or cut, the island probably either moved a couple inches or fillers added to meet the wall. Flooring underneath or not sometimes makes decisions too.