Self-supervised learning (SSL) has pushed speaker verification accuracy close to state-of-the-art levels, but the Transformer backbones used in most SSL encoders hinder on-device and real-time deployment. Prior compression work trims layer depth or width yet still inherits the quadratic cost of self-attention. We propose SV-Mixer, the first fully MLP-based student encoder for SSL distillation. SV-Mixer replaces Transformer with three lightweight modules: Multi-Scale Mixing for multi-resolution temporal features, Local-Global Mixing for frame-to-utterance context, and Group Channel Mixing for spectral subspaces. Distilled from WavLM, SV-Mixer outperforms a Transformer student by 14.6% while cutting parameters and GMACs by over half, and at 75% compression, it closely matches the teacher's performance. Our results show that attention-free SSL students can deliver teacher-level accuracy with hardware-friendly footprints, opening the door to robust on-device speaker verification.