  We present a theoretical framework to tackle quantum non-Markovian dynamics
based on a microscopic collision model (CM), where the bath consists of a large
collection of initially uncorrelated ancillas. Unlike standard memoryless CMs,
we endow the bath with memory by introducing inter-ancillary collisions between
next system-ancilla interactions. Our model interpolates between a fully
Markovian dynamics and the continuous interaction of the system with a single
ancilla, i.e., a strongly non-Markovian process. We show that in the continuos
limit one can derive a general master equation, which while keeping such
features is guaranteed to describe an unconditionally completely positive and
trace-preserving dynamics. We apply our theory to an atom in a dissipative
cavity for a Lorentzian spectral density of bath modes, a dynamics which can be
exactly solved. The predicted evolution shows a significant improvement in
approaching the exact solution with respect to two well-known memory-kernel
master equations.
