Rich information is provided by 360-degree videos. However, non-uniform geometric deformation caused by sphere-to-plane projection significantly decreases tracking accuracy of existing trackers, and the huge amount of data makes it difficult to achieve real-time tracking. Thus, this paper proposes a Siamese networks-based people tracker using template update for 360-degree equi-angular cubemap (EAC) format videos. Face stitching overcomes the problem of content discontinuity of the EAC format and avoids raising new geometric deformation in stitched images. Fully convolutional Siamese networks enable tracking at high speed. Mostly important, to be robust against combination of non-uniform geometric deformation of the EAC format and partial occlusions caused by zero padding in stitched images, this paper proposes a novel Bayes classifier-based timing detector of template update by referring to the linear discriminant feature and statistics of a score map generated by Siamese networks. Experimental results show that the proposed scheme significantly improves tracking accuracy of the fully convolutional Siamese networks SiamFC on the EAC format with operation beyond the frame acquisition rate. Moreover, the proposed score map-based timing detector of template update outperforms state-of-the-art score map-based timing detectors.