The numbers are millimeters.

"c" isn't referring to a unit of measure, it's just part of the old French rim size system.
The number (700) refers roughly to the inflated diameter of a wheel in millimeters, and the letter (c) refers to tire width. A 700a rim would be a very wide-diameter rim with a narrow tire, a 700b rim would be lower-diameter and would be fitted with a wider tire, etc.
Today, most of the old French sizes are obsolete, and nobody cares about matching that 700mm inflated diameter. So, "700c" now just refers to a rim diameter. In particular, a 700c rim is a rim with a bead seat diameter of 622mm.

A correct description for a 35mm tire on a 700c rim would be 700c x 35mm. Or, in a perfect world where we just drop the nonsense and use ISO sizes, we could call it 35 - 622.

Moving the c to the tire width ("700x35c") is abbreviation slang. And because 650c isn't currently very popular, people sometimes drop explicit references of the tire diameter entirely, and just say "35c." Which, when you think about it, is a remarkably indirect way to describe a tire's dimensions.

Unfortunately, I'm not sure I've ever seen someone try to append a "c" to an imperial tire width measurement, and just use that to describe tire size.