Objective: An electroencephalogram (EEG)-based brain-computer interface (BCI) is a direct communication pathway between the human brain and a computer. Most research so far studied more accurate BCIs, but much less attention has been paid to the ethics of BCIs. Aside from task-specific information, EEG signals also contain rich private information, e.g., user identity, emotion, disorders, etc., which should be protected. Approach: We show for the first time that adding user-wise perturbations can make identity information in EEG unlearnable. We propose four types of user-wise privacy-preserving perturbations, i.e., random noise, synthetic noise, error minimization noise, and error maximization noise. After adding the proposed perturbations to EEG training data, the user identity information in the data becomes unlearnable, while the BCI task information remains unaffected. Main results: Experiments on six EEG datasets using three neural network classifiers and various traditional machine learning models demonstrated the robustness and practicability of the proposed perturbations. Significance: Our research shows the feasibility of hiding user identity information in EEG data without impacting the primary BCI task information.