Such a group has been found by N. Dunfield, see the appendix to this preprint. The group is the fundamental group of a compact hyperbolic three--manifold which has injectivity radius large enough so that it is known to have unique products (and a little more) by a result of Delzant--Bowditch, but Nathan checked "by hand" that it is not left-orderable (by the same method as in his Inventiones paper with D. Calegari, which you should check out if you want more examples of non-left/right-orderable groups).

Such a group has been found by N. Dunfield, see the appendix to the paper

• Steffen Kionke, Jean Raimbault, Nathan Dunfield, On geometric aspects of diffuse groups, Documenta Mathematica, Vol. 21 (2016), 873-915, journal, arXiv:1411.6449.

The group is the fundamental group of a compact hyperbolic three--manifold which has injectivity radius large enough so that it is known to have unique products (and a little more) by a result of Delzant--Bowditch, but Nathan checked "by hand" that it is not left-orderable (by the same method as in his Inventiones paper with D. Calegari,

• Danny Calegari, Nathan M. Dunfield, Laminations and groups of homeomorphisms of the circle, Invent. Math. 152 (2003) 149-207, doi:10.1007/s00222-002-0271-6, arXiv:math/0203192

which you should check out if you want more examples of non-left/right-orderable groups).