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We consider a fluid-structure interaction problem with Navier-slip boundary conditions in which the fluid is considered as a non-Newtonian fluid and the structure is described by a nonlinear multi-layered model. The fluid domain is driven by a nonlinear elastic shell and thus is not fixed. To simplify the problem, we map the moving fluid domain into a fixed domain by applying an arbitrary Lagrange Euler mapping. Unlike the classical method by which we can consider the problem as its entirety, we utilize the time-discretization and split the problem into a fluid subproblem and a structure subproblem by an operator splitting scheme. Since the structure subproblem is nonlinear, Lax-Milgram lemma does not hold. Here we prove the existence and uniqueness by means of the traditional semigroup theory. Noticing that the Non-Newtonian fluid possesses a $ p- $Laplacian structure, we show the existence and uniqueness of solutions to the fluid subproblem by considering the Browder-Minty theorem. With the uniform energy estimates, we deduce the weak and weak* convergence respectively. By a generalized Aubin-Lions-Simon Lemma proposed by Muha and Canić [J. Differential Equations {\bf 266} (2019), 8370--8418], we obtain the strong convergence. Finally, we construct the test functions and pass the approximate weak formulation to the limit as time step goes to zero with the convergence results.