A plane
A flat surface that extends forever is a plane
A plane.
A plane
A plane
Plane
A plane.
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 plane
It does not have a flat surface.
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).
A flat, level surface is a plane.