  We show that for a $C^1$-open and $C^{r}$-dense subset of the set of ergodic
iterated function systems of conservative diffeomorphisms of a finite-volume
manifold of dimension $d\geq 2$, the extremal Lyapunov exponents do not vanish.
In particular, the set of non-uniform hyperbolic systems contains a $C^1$-open
and $C^r$-dense subset of ergodic random products of i.i.d. conservative
surface diffeomorphisms.
