Abstract. Turbulence models predict low droplet-collision rates in stratocumulus clouds, which should imply a narrow droplet-size distribution and little rain. Contrary to this expectation, rain is often observed in stratocumulus. In this paper we explore the hypothesis that some droplets can grow well above the average, because small-scale turbulence allows them to reside at cloud top for a time longer than the convective-eddy time t*. Long-resident droplets can grow larger because condensation due to long-wave radiative cooling and collisions have more time to enhance droplet growth. We investigate the trajectories of one billion Lagrangian droplets in direct numerical simulations of a cloudy mixed-layer configuration that is based on observations from the flight 11 from the VERDI campaign. High resolution is employed to represent a well-developed turbulent state at cloud top. Only one-way coupling is considered. We observe that 70% of the droplets spend less than 0.6t* at cloud top before leaving the cloud, while 15% of the droplets remain at least 0.9t* at cloud top. Besides, 0.2% of the droplets spend more than 2.5t* at cloud top and decouple from the large-scale convective eddies that brought them to the top, with the result that they become memoryless. Modeling collisions like a Poisson process leads to the conclusion that most rain droplets originate from those memoryless droplets. Furthermore, most long-resident droplets accumulate at the downdraft regions of the flow, which could be related to the closed-cell stratocumulus pattern. Finally, we see that condensation due to long-wave radiative cooling considerably broadens the cloud-top droplet-size distribution: 6.5% of the droplets double their mass due to radiation in their time at cloud top. This simulated droplet size distribution matches the flight measurements, confirming that condensation due to long-wave radiation can be an important mechanism for broadening the droplet-size-distribution in radiatively-driven stratocumulus.