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Most existing hyperspectral image super-resolution methods require modifications for different scales, limiting their flexibility in arbitrary-scale reconstruction. 2D Gaussian splatting provides a continuous representation that is compatible with arbitrary-scale super-resolution. Existing methods often rely on rasterization strategies, which may limit flexible spatial modeling. Extending them to hyperspectral image super-resolution remains challenging, as the task requires adaptive spatial reconstruction while preserving spectral fidelity. This paper proposes GaussianHSI, a Gaussian-Splatting-based framework for arbitrary-scale hyperspectral image super-resolution. We develop a Voronoi-Guided Bilateral 2D Gaussian Splatting for spatial reconstruction. After predicting a set of Gaussian functions to represent the input, it associates each target pixel with relevant Gaussian functions through Voronoi-guided selection. The target pixel is then reconstructed by aggregating the selected Gaussian functions with reference-aware bilateral weighting, which considers both geometric relevance and consistency with low-resolution features. We further introduce a Spectral Detail Enhancement module to improve spectral reconstruction. Extensive experiments on benchmark datasets demonstrate the effectiveness of GaussianHSI over state-of-the-art methods for arbitrary-scale hyperspectral image super-resolution.