The use of drones for last-mile parcel delivery has attracted popularity of attention, especially among online retailers and food delivery platforms that pursuit a fast and high-quality one-to-one service. However, previous works disregarded or assumed the load capacity of drones as fixed values for simplification in truck-drone delivery, such that there is no split delivery assignments on drones. Yet in real-world applications, considering drone endurance and payload capacity, it is highly practical that they deliver in a split way. This paper concerns the novel variant: the split delivery vehicle routing problem with drones (VRPD-S). We deal with the cases where each truck is equipped with a single drone, for serving a set of customers with minimal delivery duration. This problem is challenging since drones can perform multiple deliveries for a single customer node, i.e., the customers' demands are allowed to be split. Whereas trucks follow the vehicle routing problem and meanwhile need to synchronize with drones at the depot or demand points. We model this problem as a mixed-integer linear program and implement a problem-guided predictive search combined with an adaptive large neighborhood search (PS-ALNS) procedure to solve it. Extensive computational experiments were carried out on benchmark instances available in the literature. According to the numerical results, our model is validated and the proposed algorithm works effectively in small and larger instances. Split deliveries can generate savings both in terms of number of vehicles and travel costs, and lead to more flexible and efficient results for the overall delivery system.