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In this work, we explore the implications of the Cohen and Glashow Very Special Relativity (VSR) theory, a framework that introduces Lorentz invariance violation through the presence of a preferred direction. Our analysis focuses on the impact of VSR on the Cherenkov angle, revealing modifications to the dispersion relation of particles, particularly the photon and the electron, which acquire an effective inertial mass. This modification also implies a deviation in the speed of light, which can be constrained through precise experimental measurements. Using data from the RICH system of the LHCb experiment, we take advantage of its capability to reconstruct Cherenkov angles within the momentum range of the particles of 2.6-100 GeV/c. These measurements, combined with the most stringent laboratory tests of the isotropy of the speed of light ($\Delta c / c \sim 10^{-17}$), allow us to impose new upper bounds on the parameter $\Omega$, which quantifies a deviation from the standard Special Relativity. Furthermore, we establish an analogy between VSR and Minkowski's electrodynamics in a dielectric medium for particles with very high velocity, offering a physically intuitive interpretation of the parameter $\Omega$., Comment: 23 pages, 9 pages