Providing protected access to many everyday-used resources is becoming increasingly necessary. Research on applying eye movement for this purpose has been conducted for many years. However, due to technological advancements and the lack of stable solutions, subsequent explorations remain valid. The presented work is one of such studies. Two methods of biometric identification based on eye movements that utilize neural networks have been developed. In the first case, a feature vector was constructed from a 100-element time series depicting eye movement dynamics, which included velocity, acceleration, jerk, their point-to-point percentage changes, and frequency-domain representations. The same eye movement dynamic features were used in the second method, but this time, statistical values were calculated based on the previously defined time series. Long Short-Term Memory (LSTM) and dense networks were used in the user identification task in the first and second approaches, respectively. In the exploration, the publicly available GazeBase dataset was used, from which data collected for the 'jumping point' stimulus were chosen. The obtained results are very promising, with an accuracy of 96% for the LSTM model and the time series feature vector set and 76% for the second method. They were achieved over a three-year time span of eye movement recordings; however, different time periods were investigated, as well as various numbers of stimulus positions.