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For every $c\geq 1$, we define a strengthening of Kazhdan's Property (T) byconsidering uniformly bounded representations $\pi$ with fixed bound $|\pi|\leqc$. We carry out a systematic study of this property and show that it can becharacterised by the weak*-continuity of the unique invariant mean on asuitable space of coefficients. For countable groups, we prove that the familyof properties thus obtained yield an invariant at the von Neumann algebralevel. Moreover, by focusing on certain representations of rank 1 Lie groups,we show that $\operatorname{Sp}(n,1)$ and $F_{4,-20}$ admit proper uniformlyLipschitz affine actions on Hilbert spaces.

Property (T) for uniformly bounded representations and weak*-continuity of invariant means