Visual Place Recognition (VPR) localizes a query image by matching it against a database of geo-tagged reference images, making it essential for navigation and mapping in robotics. Although Vision Transformer (ViT) solutions deliver high accuracy, their large models often exceed the memory and compute budgets of resource-constrained platforms such as drones and mobile robots. To address this issue, we propose TeTRA, a ternary transformer approach that progressively quantizes the ViT backbone to 2-bit precision and binarizes its final embedding layer, offering substantial reductions in model size and latency. A carefully designed progressive distillation strategy preserves the representational power of a full-precision teacher, allowing TeTRA to retain or even surpass the accuracy of uncompressed convolutional counterparts, despite using fewer resources. Experiments on standard VPR benchmarks demonstrate that TeTRA reduces memory consumption by up to 69% compared to efficient baselines, while lowering inference latency by 35%, with either no loss or a slight improvement in recall@1. These gains enable high-accuracy VPR on power-constrained, memory-limited robotic platforms, making TeTRA an appealing solution for real-world deployment.