Vision Mamba has recently received attention as an alternative to Vision Transformers (ViTs) for image classification. The network size of Vision Mamba scales linearly with input image resolution, whereas ViTs scale quadratically, a feature that improves computational and memory efficiency. Moreover, Vision Mamba requires a significantly smaller number of trainable parameters than traditional convolutional neural networks (CNNs), and thus, they can be more memory efficient. Because of these features, we introduce, for the first time, a neural network that uses Vision Mamba as its backbone for predicting the permeability of three-dimensional porous media. We compare the performance of Vision Mamba with ViT and CNN models across multiple aspects of permeability prediction and perform an ablation study to assess the effects of its components on accuracy. We demonstrate in practice the aforementioned advantages of Vision Mamba over ViTs and CNNs in the permeability prediction of three-dimensional porous media. We make the source code publicly available to facilitate reproducibility and to enable other researchers to build on and extend this work. We believe the proposed framework has the potential to be integrated into large vision models in which Vision Mamba is used instead of ViTs.