In integrated ground-air-space (IGAS) wireless networks, numerous services require sensing knowledge including location, angle, distance information, etc., which usually can be acquired during the beam training stage. On the other hand, IGAS networks employ large-scale antenna arrays to mitigate obstacle occlusion and path loss. However, large-scale arrays generate pencil-shaped beams, which necessitate a higher number of training beams to cover the desired space. These factors motivate our investigation into the IGAS beam training problem to achieve effective sensing services. To address the high complexity and low identification accuracy of existing beam training techniques, we propose an efficient hashing multi-arm beam (HMB) training scheme. Specifically, we first construct an IGAS single-beam training codebook for the uniform planar arrays. Then, the hash functions are chosen independently to construct the multi-arm beam training codebooks for each AP. All APs traverse the predefined multi-arm beam training codeword simultaneously and the multi-AP superimposed signals at the user are recorded. Finally, the soft decision and voting methods are applied to obtain the correctly aligned beams only based on the signal powers. In addition, we logically prove that the traversal complexity is at the logarithmic level. Simulation results show that our proposed IGAS HMB training method can achieve 96.4% identification accuracy of the exhaustive beam training method and greatly reduce the training overhead.