Renormings of c0 and the fixed point property

It is unknown whether all renormings of c0 fail the fixed point property (FPP) or not. In this note we give a sufficient condition for a renorming of c0 to fail the FPP which is more general than the previously known ones. In fact the condition is satisfied for all renormings of c0 we know. It also allows us to show that for some usual renormings of c0 not only the whole space but all its infinite dimensional subspaces fail the FPP.