Recently, a large class of quantum non-Markovian piecewise dynamics for an
open quantum system obeying closed evolution equations has been introduced [1]. These dynamics have been defined in terms of a waiting-time distribution between quantum jumps, along with quantum maps describing the effect of jumps and the system evolution between them. Here, we present a quantum collision model with memory, whose reduced dynamics in the continuous-time limit reproduces the above class of non-Markovian piecewise dynamics, thus providing an explicit microscopic realization.