Unique product group which is not right orderable

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).

Every U.P. group is a t.u.p group. See

Andrzej Strojnowski, A note on u.p. groups, Communications in Algebra, 8:3, (1980) 231-234.