We report experiments concerning the deposition of a droplet on a solid surface and the related spontaneous spreading, during which a secondary droplet is ejected. Experimental investigations and theoretical considerations have been performed in order to understand the reasons of the formation of this droplet and of its ejection and to estimate the conditions that induce these phenomena. High-speed imaging and specific deposition conditions have been necessary to visualize such phenomena. It has been shown that the ejection is possible in the complete wetting regime when low impact inertia and high position of the center of mass of the drop before spreading are achieved simultaneously. A model taking into account non-stationarity, inertia, wetting capillarity and viscous effects has been developed. It resulted in two dimensionless numbers Pi(1) and Pi(2) that characterize the occurrence of the ejection phenomenon.