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Galaxy clusters detected through the thermal Sunyaev-Zeldovich (tSZ) effect are a powerful cosmological probe from which constraints on cosmological parameters such as $Ω_{\mathrm{m}}$ and $σ_8$ can be derived. The measured cluster tSZ signal can be, however, contaminated by Cosmic Infrared Background (CIB) emission, as the CIB is spatially correlated with the cluster tSZ field. We quantify the extent of this contamination by applying the iterative multi-frequency matched filter (iMMF) cluster-finding method to mock Planck-like data from the Websky simulation. We find a significant bias in the retrieved cluster tSZ observables (signal-to-noise and Compton-$y$ amplitude), at the level of about $0.5\, σ$ per cluster. This CIB-induced bias translates into about $20$% fewer detections than expected if all the Planck HFI channels are used in the analysis, which can potentially bias derived cosmological constraints. We introduce a spectrally constrained iMMF, or sciMMF, which proves to be highly effective at suppressing this CIB-induced bias from the tSZ cluster observables by spectrally deprojecting the cluster-correlated CIB at the expense of a small signal-to-noise penalty. Our sciMMF is also robust to modelling uncertainties, namely to the choice of deprojection spectral energy distribution. With it, CIB-free cluster catalogues can be constructed and used for cosmological inference. We provide a publicly available implementation of our sciMMF as part of the SZiFi package.