Examples of non-cubulated hyperbolic groups

If a group $G$ satisfies Kazhdan's property (T), then any action of $G$ on a CAT(0) cube complex has a global fixed point. See Niblo and Roller's article Groups acting on cubes and Kazhdan's Property (T). Examples of hyperbolic groups which satisfy this property include:

• Uniform lattices in quaternionic hyperbolic spaces.
• Random groups in Gromov's model for some density.

Also, by a theorem due to Gromov, any hyperbolic group admits a quotient which is hyperbolic and has property (T).

As @AGenevois says in his answer, the standard examples of non-cubulated hyperbolic groups are those with Kazhdan's property (T), such as quaternionic hyperbolic lattices.

Complex hyperbolic lattices (in dimension >2) provide a more delicate class of examples. On the one hand, they are not cubulable, by a theorem of Delzant--Py. On the other hand, they are known to have the Haagerup property (I think this is proved in the book by Bekka--de la Harpe--Valette), so they also don't have property (T).

As far as I know, they are the only class of examples of hyperbolic groups known to be Haagerup but non-cubulable. I'd be interested to hear of others.