A locally compact group G has Kazhdan\'s property (T) if every action of G by affine isometries on a Hilbert space has a fixed point.

In this talk we will be interested in strengthening this property by replacing the Hilbert space by other Banach spaces. In particular, we will show how to generalize the geometric/spectral method for proving property (T) to the setting of reflexive Banach spaces. We will also discuss examples and present several applications.