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](https://www.math.uni-bielefeld.de/documenta/vol-21/24.html), arXiv:[1411.6449](https://arxiv.org/abs/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](https://doi.org/10.1007/s00222-002-0271-6), arXiv:[math/0203192](https://arxiv.org/abs/math/0203192)

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