Answer
The statement is false, because in space, it is called a bisector plane. Because in space the locus of points that are equidistant from both ends of the line segment, is a plane.
Work Step by Step
In a plane, the perpendicular bisector of a line segment is unique. In space, it is called a bisector plane. Because in space the locus of points that are equidistant from both ends of the line segment, is a plane. The plane is constructed of all the perpendicular bisectors of $AB$. We can imagine this, by drawing one of them and rotating it around the midpoint of $AB$.
