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Non-Fermi-liquid (NFL), a significant deviation from Fermi-liquid theory, usually emerges near an order-disorder phase transition at absolute zero. Recently, a diverging susceptibility toward zero temperature was observed in a quasicrystal (QC). Since an electronic long-range ordering is normally absent in QCs, this anomalous behaviour should be a new type of NFL. Here we study high-resolution partial-fluorescence-yield x-ray absorption spectroscopy on Yb-based intermediate-valence icosahedral QCs and cubic approximant crystals (ACs), some of which are new materials, to unveil the mechanism of the NFL. We find that for both forms of QCs and ACs, there is a critical lattice parameter where Yb-valence and magnetism concomitantly exhibit singularities, suggesting a critical-valence-fluctuation-induced NFL. The present result provides an intriguing structure-property relationship of matter; size of a Tsai-type cluster (that is a common local structure to both forms) tunes the NFL whereas translational symmetry (that is present in ACs but absent in QCs) determines the nature of the NFL against the external/chemical pressure.