Without seeing the rest of your kitchen it's hard to say, but I'd go with the same beadboard that's on the ceiling painted as you suggest.

I think I would just paint it the wall color rather than separate it out with a different treatment.

How about a fun patterned wallpaper in a color you like?

I suspect tile would be a lot easier to swipe clean with a damp cloth, and even scrub if necessary - wouldn't the wallpaper get grunge stuck in those little grooves over time, and be very difficult to clean?

Bead board wallpaper is better option as it is easy to install.

I basically agree, although depending in the rest of the room, making this a little focal niche with paint or tile might be quite charming. You can have fun with it to see how you like it.

What strikes me is that bead board, which is a rather humble material, doesn't really go with the more elegant, front-parlor door style. It can look right on the ceiling, as if it were original to the plainer kitchens of the past and left there as the room continued to evolve to today, but I wouldn't deliberately spot it around. If it were on the wall originally, it'd have been floor to ceiling or partial height on much or most of the room.