Flying networks, composed of Unmanned Aerial Vehicles (UAVs) acting as mobile Base Stations and Access Points, have emerged to provide on-demand wireless connectivity, especially due to their positioning capability. Still, existing solutions are focused on improving aggregate network performance using a best-effort approach. This may compromise the use of multiple services with different performance requirements. Network slicing has emerged in 5G networks to address the problem, allowing to meet different Quality of Service (QoS) levels on top of a shared physical network infrastructure. However, Mobile Network Operators typically use fixed Base Stations to satisfy the requirements of different network slices, which may not be feasible due to limited resources and the dynamism of some scenarios. We propose an algorithm for enabling the joint placement and allocation of communications resources in Slicing-aware Flying Access and Backhaul networks – SurFABle. SurFABle allows the computation of the amount of communications resources needed, namely the number of UAVs acting as Flying Access Points and Flying Gateways, and their placement. The performance evaluation carried out by means of ns-3 simulations and an experimental testbed shows that SurFABle makes it possible to meet heterogeneous QoS levels of multiple network slices using the minimum number of UAVs.