  We give a method to construct Poisson brackets $\{\cdot,\cdot\}$ on Banach
manifolds~$M$, for which the value of $\{f,g\}$ at some point $m\in M$ may
depend on higher order derivatives of the smooth functions $f,g\colon
M\to{\mathbb R}$, and not only on the first-order derivatives, as it is the
case on all finite-dimensional manifolds. We discuss specific examples in this
connection, as well as the impact on the earlier research on Poisson geometry
of Banach manifolds. Those brackets are counterexamples to the claim that the
Leibniz property for any Poisson bracket on a Banach manifold would imply the
existence of a Poisson tensor for that bracket.
