You are probably looking for the word 'plane'.
The flat surface that goes on and on in all directions has no boundaries. This flat surface is called infinite surface. Accordingly, the answer is that the flat surface that goes on and on in all directions is called an infinite surface.
A plane
It is a plane.
Plane
Plane
a plane
A plane could be considered an endless flat surface. A plane goes out infinitely in all directions, therefore, creating an endless flat surface.
plane
It is a plane.
There is no real world flat surface which goes on forever. It is a geometric idealised concept. So, since there is not even a first such thing, there cannot be another.
A flat surface, like a sheet of paper, with no edges (ie: it goes on and on and on and on....) or boundaries. What you're describing is a 2 dimensional plane. No, not plane like an airplane. A plane is simply a flat "sheet" that goes on forever in 2 dimensions: left/right (wide) and up/down (tall).
One curved surface (that goes round the cylinder) and 2 flat ends