Accurate simulation of flexural wave propagation in unbounded domains requires effective boundary truncation to suppress artificial reflections. This paper presents a time-domain, unsplit-field perfectly matched layer (PML) formulation for unbounded Euler-Bernoulli beams (EB-PML). The proposed method is derived by introducing complex coordinate stretching into the frequency-domain governing equations to induce wave attenuation, which are subsequently transformed into the time domain via the inverse Fourier transform. This formulation yields a system of equations expressed in terms of beam deflection and bending moment. A mixed-like finite element framework is adopted, in which deflection and bending moment are treated as independent primary variables. This approach enables a stable time-domain implementation without the use of auxiliary variables, thereby overcoming the limitations associated with conventional high-order derivative formulations. For numerical implementation, the variational forms are spatially discretized using cubic Hermite shape functions, resulting in semi-discrete equations that are integrated in time using a third-order Newmark-β scheme. The performance of the proposed EB-PML is rigorously validated against exact analytical solutions in both the time and frequency domains. The time-domain verification examines the transient response of an infinite beam subjected to a unit-impulse load, demonstrating excellent agreement between numerical and exact solutions. Frequency-domain validation is conducted under harmonic excitation, where steady-state responses are compared with exact complex-valued amplitudes. Quantitative error analysis shows that the EB-PML effectively suppresses boundary reflections, with normalized L2-errors remaining below 0.02% in both domains. The proposed framework is well suited for practical applications, including wave-based damage assessment of lifeline structures such as water and gas pipelines.