Nonlinear rheological features were investigated for an aqueous solution of tetraethylammonium perfluorooctyl sulfonate (C8F17SO3-N+(C2H5)(4); abbreviated as FOSTEA). In the solution (c=0.045 mol/l; 28.3 g/l), spherical micelles of FOSTEA were connected with each other to form threads of pearl-necklace shape. These threads were further organized into a transient network to exhibit linear relaxation characteristic of living polymers, single-mode terminal relaxation widely separated from faster relaxation processes. Nonlinear relaxation experiments against large step-strains gamma(less than or equal to 8) revealed that the terminal relaxation mode had a gamma-insensitive relaxation time but its relaxation intensity exhibited significant damping (much stronger than that for entangled polymers). In contrast, the relaxation time and intensity for the fast relaxation modes first increased and then decreased with increasing gamma. Under shear flow, the FOSTEA threads exhibited strong thinning of the viscosity. These nonlinear features of the FOSTEA threads were compared with those of other threadlike micelles, analyzed on the basis of an empirically introduced constitutive equation, and discussed in relation to strain/low-induced scission of the living threads.