Conventional model predictive current control of permanent magnet synchronous machines (PMSMs) relies heavily on a precise mathematical model, which may be challenging to obtain in certain cases. To address this issue, this article proposes a model-free predictive current controller for PMSMs. Specifically, the dynamic model of the motor currents is represented using two ultralocal models, which express the derivatives of a controlled output as the sum of an amplified control input and an unknown offset term. All system parameters, nonlinear terms, and unmodeled dynamics are encapsulated into two offset terms. Online estimation of these two offset terms is performed using a second-order fixed-time convergence observer without requiring exact knowledge of the system parameters in advance. In contrast to both the linear extended state observer and the super-twisting observer, the fixed-time observer enhances convergence speed while concurrently sustaining minimal chatting. To get rid of the need for model rated parameters, an extremum-seeking approach is utilized to tune the control gains in the ultralocal model, where the amplitude of the disturbance item in extremum-seeking approach is adaptively attenuated, resulting in effective reduction of the fluctuations of both the control gains and PMSM's currents in steady state. Subsequently, a predictive controller is designed using this ultralocal model. Finally, the effectiveness and advantages of the proposed control scheme are confirmed through experimental results.