Since the hydraulic axial piston pump is the engine that drives hydraulic transmission systems, it is widely utilized in aerospace, marine equipment, civil engineering, and mechanical engineering. Operating safely and dependably is crucial, and failure poses a major risk. Hydraulic axial piston pump malfunctions are characterized by internal concealment, challenging self-adaptive feature extraction, and blatant timing of fault signals. By completely integrating the time-frequency feature conversion capability of synchrosqueezing wavelet transform (SWT), the feature extraction capability of VGG11, as well as the feature memory capability of the long short-term memory (LSTM) model, a novel intelligent fault identification method is proposed in this paper. First, the status data are transformed into two dimensions in terms of time and frequency by using SWT. Second, the depth features of the time–frequency map are obtained and dimensionality reduction is carried out by using the deep feature mining capability of VGG11. Third, LSTM is added to provide the damage identification model for long-term memory capabilities. The Softmax layer is utilized for the intelligent evaluation of various damage patterns and health state. The proposed method is utilized to identify and diagnose five typical states, including normal state, swash plate wear, sliding slipper wear, loose slipper, and center spring failure, based on the externally observed vibration signals of a hydraulic axial piston pump. The results indicate that the average test accuracy for five typical state signals reaches 99.43%, the standard deviation is 0.0011, and the average test duration is 2.675 s. The integrated model exhibits improved all-around performance when compared to LSTM, LeNet-5, AlexNet, VGG11, and other typical models. The proposed method is validated to be efficient and accurate for the intelligent identification of common defects of hydraulic axial piston pumps.