BPSK modulated GPS L1 CA signal is the most widely used GNSS signal to date, and the first path detection (FPD) of the conventional GPS L1 CA signals is the most chal- lenging problem to ensure reliable GPS positioning in multipath environments. In this paper, we propose an FPD network (FPDN) based on multi-layer perceptron (MLP)-Mixer to extract the first path from the discrete autocorrelation function (ACF) output accurately with low computational cost. In addition, the proposed FPDN is useful in practice because it is robust to noise and achieves a high FPD performance without any prior assumption on the number of total incoming multipath, which is required for conventional signal processing-based FPD techniques. We compare the performance of the proposed FPDN to that of diverse conventional techniques, such as techniques based on narrow correlator, super-resolution, and some widely used CNNs such as VGGNet, ResNet, and U-Net, through simulations and field tests. As demonstrated, the proposed FPDN outperforms all of the compared FPD techniques in terms of the computational cost and accuracy for wide range of carrier-to-noise (C/N0) ratios.