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We consider a general class of stochastic optimal control problems, where thestate process lives in a real separable Hilbert space and is driven by acylindrical Brownian motion and a Poisson random measure; no special structureis imposed on the coefficients, which are also allowed to be path-dependent; inaddition, the diffusion coefficient can be degenerate. For such a class ofstochastic control problems, we prove, by means of purely probabilistictechniques based on the so-called randomization method, that the value of thecontrol problem admits a probabilistic representation formula (known asnon-linear Feynman-Kac formula) in terms of a suitable backward stochasticdifferential equation. This probabilistic representation considerably extendscurrent results in the literature on the infinite-dimensional case, and it isalso relevant in finite dimension. Such a representation allows to show, in thenon-path-dependent (or Markovian) case, that the value function satisfies theso-called randomized dynamic programming principle. As a consequence, we areable to prove that the value function is a viscosity solution of thecorresponding Hamilton-Jacobi-Bellman equation, which turns out to be asecond-order fully non-linear integro-differential equation in Hilbert space.

BSDE Representation and Randomized Dynamic Programming Principle for Stochastic Control Problems of Infinite-Dimensional Jump-Diffusions