Algorithms for answering the k nearest-neighbor (k-NN) query are widely used for queries in spatial databases and for distance classification of a group of query points against a reference dataset to derive the dominating feature class. GPU devices have significantly more processing cores than CPUs and faster device memory than the main memory accessed by CPUs, thus, providing higher computing power for processing demanding queries like the k-NN. However, since device and/or main memory may not be able to host an entire, rather big, reference and query datasets, storing these datasets in a fast secondary device, like a solid state disk (SSD), and partially retrieve the required, at each stage, partitions is, in many practical cases, a feasible solution. We propose and implement the first GPU-based algorithms for processing the k-NN query for big reference and query spatial data stored on SSDs. Based on 3d synthetic and real big spatial data, we experimentally compare these algorithms and highlight the most efficient algorithmic variation. This variation utilizes a CUDA feature known as Concurrent Kernel Execution, to further improve its performance.