Until recently, I thought that any theory that contains non-renormalizable interactions in the power-counting sense (i.e. those whose couplings have negative mass dimension) must be an 'effective' theory that necessarily breaks down at some energy. However, I've been looking at Weinberg's QFT textbook and according to him it might be possible for a perturabatively non-renormalizable theory to be asymptotically safe. Does anyone know if this is just a conjecture, or whether there actually exist successful and convincing physical models of a perturbatively non-renormalizable theory that is asymptotically safe? Any info would be most appreciated.