The most well-understood examples are the ones you mention: Axiom A diffeomorphisms and Markov maps of the interval, since these can be modeled by SFTs. Note that "Bernoulli" refers to a particular choice of invariant measure; the SRB measure for an Axiom A attractor (or the ACIP for a Markov interval map) is Bernoulli, but other invariant measures, such as periodic orbit measures, need not be. More generally, given a system modeled by an SFT, equilibrium states for Hölder continuous potentials are always Bernoulli. (Recall that these are invariant measures $\mu$ that maximize the quantity $h(\mu) + \int\phi\,d\mu$, where $h(\mu)$ is Kolmogorov-Sinai entropy and $\phi\colon X\to \mathbb{R}$ is the potential function.)

Beyond this, I know of basically two classes of examples that are known to be Bernoulli; both are "non-uniformly hyperbolic" in some sense. The first class contains systems that can be modeled by a Young tower or a countable-state Markov shift, where the measure is an equilibrium state for some sufficiently regular potential function; this includes the case where the measure is SRB, in particular when the measure is smooth. For example, this includes any positive entropy equilibrium state for a Hölder continuous potential on a surface diffeomorphism, thanks to recent work of Omri Sarig (JAMS 2013, JMD 2011).

In fact, if the measure is smooth and has non-zero Lyapunov exponents, then you don't need to build a Young tower or a countable-state Markov shift; for measures like this ("hyperbolic" measures), Bernoullicity was proved by Yakov Pesin in 1977. In particular, this includes Liouville measure for geodesic flow of a compact surface of genus at least two without focal points (of course this is continuous-time rather than discrete-time as you asked for). The result for smooth invariant measures was extended to SRB measures by Ledrappier in 1984 (he also did ACIPs for interval maps without any Markov assumption).

notBernoulli?". I think the answer might be "none". $\endgroup$