Recent speaker verification (SV) systems have shown a trend toward adopting deeper speaker embedding extractors. Although deeper and larger neural networks can significantly improve performance, their substantial memory requirements hinder training on consumer GPUs. In this paper, we explore a memory-efficient training strategy for deep speaker embedding learning in resource-constrained scenarios. Firstly, we conduct a systematic analysis of GPU memory allocation during SV system training. Empirical observations show that activations and optimizer states are the main sources of memory consumption. For activations, we design two types of reversible neural networks which eliminate the need to store intermediate activations during back-propagation, thereby significantly reducing memory usage without performance loss. For optimizer states, we introduce a dynamic quantization approach that replaces the original 32-bit floating-point values with a dynamic tree-based 8-bit data type. Experimental results on VoxCeleb demonstrate that the reversible variants of ResNets and DF-ResNets can perform training without the need to cache activations in GPU memory. In addition, the 8-bit versions of SGD and Adam save 75% of memory costs while maintaining performance compared to their 32-bit counterparts. Finally, a detailed comparison of memory usage and performance indicates that our proposed models achieve up to 16.2x memory savings, with nearly identical parameters and performance compared to the vanilla systems. In contrast to the previous need for multiple high-end GPUs such as the A100, we can effectively train deep speaker embedding extractors with just one or two consumer-level 2080Ti GPUs.