During the process of high-power microwave breakdown, neutral air particles are rapidly ionized by high-intensity electromagnetic (EM) fields. The ionized particles form a cloud of plasma, which generate secondary EM fields and interact with the incident fields. In the meantime, the plasma also diffuses from higher density regions to the surrounding lower density regions. The ionization-diffusion mechanism in the nonlinear EM-plasma interaction results in a plasma front with an extremely high density gradient, which evolves and propagates in space and time. To capture such a high density gradient and the resulting highly localized EM fields, a numerical method with a very high spatial resolution is required. Based on the nodal discontinuous Galerkin time-domain method, a dynamic p-adaptation algorithm is proposed in this paper to capture the fast varying physics by changing the order of basis functions wherever and whenever needed. The phenomenon of a microwave streamer elongating under the excitation of a high-intensity standing wave is simulated to demonstrate the capability of the proposed method.