Recurrent event time data arise in many studies, including biomedicine, public health, marketing, and social media analysis. High-dimensional recurrent event data involving many event types and observations have become prevalent with advances in information technology. This paper proposes a semiparametric dynamic factor model for the dimension reduction of high-dimensional recurrent event data. The proposed model imposes a low-dimensional structure on the mean intensity functions of the event types while allowing for dependencies. A nearly rate-optimal smoothing-based estimator is proposed. An information criterion that consistently selects the number of factors is also developed. Simulation studies demonstrate the effectiveness of these inference tools. The proposed method is applied to grocery shopping data, for which an interpretable factor structure is obtained.