The language-specific domain adaptation problem refers to speech processing in resource-constrained embedded systems when pre-trained large models cannot be applied. This paper investigates language-specific adaptation strategies for automatic text-independent speaker recognition on an open set of speakers for various languages. MobileNetV3 was chosen as the most common model designed for edge applications and achieves a good accuracy-efficiency balance by using depth-wise separable convolutions to reduce the number of parameters and computations. The model was pre-trained in English and investigated for cross-language domain adaptation for German, French, Italian, Russian, Spanish, Dutch, and Chinese. We propose a combination of transfer learning and fine-tuning techniques to successfully adapt speaker verification models to a particular language. The proposed approach is validated using the CommonVoice cross-language dataset. The results demonstrate a notable improvement in the average EER up to 6% using fine-tuned models, with the most significant gains observed for linguistically distant languages. The study also identified performance degradation for speech samples shorter than 6 s or with fewer than 5 samples per speaker. Our work provides a scalable framework for the language-specific domain adaptation speaker verification in edge environments, balancing accuracy and resource efficiency.