Fingerprint authentication is widely used in biometrics due to its simple process, but it is vulnerable to fake fingerprints. This study proposes a patch-based fake fingerprint detection method using a fully convolutional neural network with a small number of parameters and an optimal threshold to solve the above-mentioned problem. Unlike the existing methods that classify a fingerprint as live or fake, the proposed method classifies fingerprints as fake, live, or background, so preprocessing methods such as segmentation are not needed. The proposed convolutional neural network (CNN) structure applies the Fire module of SqueezeNet, and the fewer parameters used require only 2.0 MB of memory. The network that has completed training is applied to the training data in a fully convolutional way, and the optimal threshold to distinguish fake fingerprints is determined, which is used in the final test. As a result of this study experiment, the proposed method showed an average classification error of 1.35%, demonstrating a fake fingerprint detection method using a high-performance CNN with a small number of parameters.