Closest Non-Bordering Countries

What about between Bahrain and Saudi Arabia (looks to be about 5 miles) and Bahrain and Qatar (looks like less than 2 miles from one of Bahrain's islands to Qatar).

Here is Middle Island which has Saudi customs on one side and Bahraini customs on the other - but try as I might, I just cannot find whether either country owns it, or if the border runs down its middle! Does anyone out there know?

Could North and South Korea technically not border because of the DMZ? Its a question not a statement

The quiz asks for the closest possible distance between countries, so any countries sharing a land border will be zero distance apart, and disqualified from the quiz, even if they have offshore islands separated by water.

How about United States & United Kingdom via their respective Virgin Islands?

UK territories do not form a part of the UK, even though they are under its jurisdiction (which is why Anguilla - Sint Maarten above has also been omitted).

What about Egypt to Saudi Arabia?

Just a bit too far—about 7 km. Same goes for Egypt and Jordan.

What about Panama and Colombia?

They do border each other, it's weird that they separate continents I know, but they have a land border.

That's 50 km

And here is the closest non-bordering country pair in the world! The leaf-shaped island in the middle distance is Namibia, all of the green river-bank in the foreground is Zimbabwe, and between them, the strip of dry land with a road running across it, is Botswana (just) keeping them apart.

I'm going to be extra nit-picky here though: this should really still be classified as a "water border." Namibia owns no territory on the south side of the Chobe River, thus you must cross water in order to get from Namibia to Zimbabwe at this shortest distance.
Plus if you look at the way the borders are drawn on Google Maps (admittedly not perfect), the narrowest they ever are is over water. I know that's not your criteria, but just added reference.

I would venture that the closest non-bordering countries that are joined only by land borders would probably be Monaco and Italy at just over 8km

Canada and Denmark (Greenland)?

Overseas territories don't count

Papua New Guinea and the Solomon Islands are about 3.7 km apart (Aiaisina Island and Kukuvulu Island)