Cavity magnomechanics (CMM) is an emerging field and has received much attention in the past decade. It deals with coherent couplings among microwave cavity photons, magnons and vibration phonons. So far, all previous CMM experiments have been operated in the weak-coupling regime. This considerably limits prospective various applications of the system. Here, we demonstrate the CMM system in the strong-coupling regime and observe the associated normal-mode splitting. In this regime, the mechanical oscillator is strongly coupled to a cavity-magnon polariton that is formed by strongly coupled cavity photons and magnons, and the polariton-mechanics cooperativity reaches $4\times10^3$, which is improved by three orders of magnitude than previous CMM experiments. The system is then in the triple-strong-coupling regime and the normal modes of the system are the hybridization of microwave photons, magnons and phonons. This is achieved by significantly reducing the decay rate of the polariton mode using coherent perfect absorption and the decay rate is reduced by four orders of magnitude. The work paves the way towards coherent control and measurement of the quantum states of phonons, photons and magnons, and provides a new platform for the study of rich strong-coupling effects in multipartite hybrid systems.