The first sense of the term is the geometry over a non-Archimedean ordered field, or a subset thereof.
In this geometry, there are significant differences from Euclidean geometry; in particular, there are infinitely many parallels to a straight line through a point—so the parallel postulate fails—but the sum of the angles of a triangle is still a straight angle.
[2] Intuitively, in such a space, the points on a line cannot be described by the real numbers or a subset thereof, and there exist segments of "infinite" or "infinitesimal" length.
The second sense of the term is the metric geometry over a non-Archimedean valued field,[3] or ultrametric space.
Intuitively, in such a space, distances fail to "add up" or "accumulate".