If you haven't discovered for yourself, go to any side, top, bottom view ortho to that plane and select a shallow (only a pixel or two deep in screen vertical resolution) fence left-to-right enclosing the plane. It can be a little fiddly to orient such a view if the plane isn't the default world or one parallel to it, but it helps to activate the gripper and toggle the C key on a vertex on the plane to help with visual orientation. Only those objects completely enclosed by the fence will be selected.