We report on a comprehensive study of CePd(1-x)Rh(x) (0.6 <or= x <or= 0.95) poly- and single crystals close to the ferromagnetic instability by means of low-temperature ac susceptibility, magnetization, and volume thermal expansion. The signature of ferromagnetism in this heavy-fermion system can be traced from 6.6 K in CePd down to 25 mK for x = 0.87. Despite pronounced non-Fermi-liquid effects in both specific heat and thermal expansion, the Grüneisen ratio does not diverge as T --> 0, providing evidence for the absence of a quantum critical point. Instead, a peculiar "Kondo-cluster-glass" state is found for x >or= 0.65, and the non-Fermi-liquid effects in the specific heat, ac susceptibility, and magnetization are compatible with the quantum Griffiths phase scenario.