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 that extends forever is a plane
A plane.
Plane
Plane
lever
A flat surface that extends forever is a plane
A plane
A plane.
A plane
A plane
Plane
A plane.
Plane
a plane
lever
A floor.
The difference between a "face" and a "flat surface" is found in what is being described by each term. A "face" refers either to the direction of a surface or to the simple fact of a surface as a surface. By contrast, a "flat surface" refers to the quality of the surface in question. Thus, one "face" of a building may be flat, smooth, curved, etc., while another may be similar or different.