Unmanned aerial vehicles (UAVs) need more autonomy. In light of inherent size, weight and power (SWaP) constraints, avionics with artificial intelligence implemented using neuromorphic technology offers a potential solution. We demonstrate intelligent drone control using spiking neural networks (SNNs), which can run on neuromorphic hardware. We present "BatSLAM", a modular SNN for autonomous localization, navigation, and control of UAVs. The bio-inspired algorithms are implemented using the neural modeling and simulation software Nengo, and are able to control the drone to autonomously perform a complex "house search" task in an AirSim simulated environment using only local sensor feedback. In this task, the drone is randomly placed and given a target object. The BatSLAM network localizes the drone, retrieves the target object location from memory, and then guides the drone along the most efficient path through the house to the target, avoiding obstacles and maneuvering through doors and stairways. We present benchmark results showing the BatSLAM network achieves 97.2 percent success rate in navigating to target objects in the house. To the best of our knowledge, the BatSLAM network presented here is the first in the world to carry out localization, navigation, and control in a fully spiking implementation.