We present novel (high-order) finite element schemes for the fluid-structure interaction (FSI) problem based on an arbitrary Lagrangian-Eulerian divergence-free hybridizable discontinuous Gakerkin (ALE divergence-free HDG) incompressible flow solver, a Tangential-DisplacementNormal-Normal-Stress (TDNNS) nonlinear elasticity solver, and a generalized Robin interface condition treatment. Temporal discretization is performed using the high-order backward difference formulas (BDFs). Both monolithic and strongly coupled partitioned fully discrete schemes are obtained. Numerical convergence studies are performed for the flow and elasticity solvers, and the coupled FSI solver, which verify the high-order space-time convergence of the proposed schemes. Numerical results on classical two dimensional benchmark problems also showed good performance of our proposed methods.