Summary: We construct a new class of asymptotically (A)dS black hole solutions of Einstein-Yang-Mills massive gravity in the presence of Born-Infeld nonlinear electrodynamics. The obtained solutions possess a Coulomb electric charge, massive term and a non-abelian hair as well. We calculate the conserved and thermodynamic quantities, and investigate the validity of the first law of thermodynamics. Also, we investigate thermal stability conditions by using the sign of heat capacity through canonical ensemble. Next, we consider the cosmological constant as a thermodynamical pressure and study the van der Waals like phase transition of black holes in the extended phase space thermodynamics. Our results indicate the existence of a phase transition which is affected by the parameters of theory. Finally, we consider a massless scalar perturbation in the background of asymptotically adS solutions and calculate the quasinormal modes by employing the pseudospectral method. The imaginary part of quasinormal frequencies is the time scale of a thermal state (in the conformal field theory) for the approach to thermal equilibrium.