In computer vision, the vision transformer (ViT) has increasingly superseded the convolutional neural network (CNN) for improved accuracy and robustness. However, ViT's large model sizes and high sample complexity make it difficult to train on resource-constrained edge devices. Split learning (SL) emerges as a viable solution, leveraging server-side resources to train ViTs while utilizing private data from distributed devices. However, SL requires additional information exchange for weight updates between the device and the server, which can be exposed to various attacks on private training data. To mitigate the risk of data breaches in classification tasks, inspired from the CutMix regularization, we propose a novel privacy-preserving SL framework that injects Gaussian noise into smashed data and mixes randomly chosen patches of smashed data across clients, coined DP-CutMixSL. Our analysis demonstrates that DP-CutMixSL is a differentially private (DP) mechanism that strengthens privacy protection against membership inference attacks during forward propagation. Through simulations, we show that DP-CutMixSL improves privacy protection against membership inference attacks, reconstruction attacks, and label inference attacks, while also improving accuracy compared to DP-SL and DP-MixSL.