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
plane
An endless flat surface, also known as an infinite plane, is a theoretical concept in geometry that extends indefinitely in all directions. It has the properties of being flat, meaning that at any point on the plane, the surface is perfectly level with no curvature. In mathematics, an infinite plane is often used as a simplified model for understanding concepts such as coordinate systems, vectors, and geometric shapes.
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