A well-known example of a contact manifold is $S^5$, arising from it being a circle bundle over $\mathbb{CP^2}$. This is somewhat related to the reduction of the structure group from $SO(5)$ to $SU(2)$, but all we need to know is that there is a 1-form $\eta$ on $S^5$, the contact form (which is, secretly, a connection for the circle bundle). Clearly $S^5$ cannot have a self-dual Yang–Mills connection, but it can possibly admit something more general, namely a connection $A$ satisfying the (anti-)self-dual contact instanton equation $$ \ast_g F_A = (-)\eta \wedge F_A \qquad (1) $$ depending on a choice of metric $g$. Given an ordinary instanton on $\mathbb{CP}^2$ we get by pullback an instanton on $S^5$, but since $SU(2)$-bundles on $\mathbb{CP}^2$ trivialise after pulling up to $S^5$, these are instantons on the trivial bundle. The tangent bundle of $S^5$ is nontrivial, since it is given by the clutching construction using the nontrivial class in $\pi_4(SU(2)) = \pi_4(S^3) = \mathbb{Z}/2$. It is not clear at all if there are connections on $TS^5$ that satisfy $(1)$ for any $g$, or for that matter, any combination of $g$ and $\eta$ (I guess we should ask for a contact metric structure, to keep things consistent). I asked David Baraglia recently about this, seeing as he wrote a paper [1] on these things, and he said the answer should be 'no' for the round metric (and the standard $\eta$), but didn't know for other cases. I don't understand David and Pedram's paper enough to be able to calculate the possible obstructions, or otherwise find such a thing, in either the self-dual, or the anti-self-dual, case.
To me it seems a really basic example, one that could, and should, be written down super-explicitly, much like one can write down formulas instantons on $S^4$...
[1] David Baraglia and Pedram Hekmati, Moduli spaces of contact instantons, Advances in Mathematics 294 (2016) 562-595, doi:10.1016/j.aim.2016.03.001, arXiv:1401.5140
