This paper presents the first bit error rate (BER) analysis of a pinching-antenna (PA)-based non-orthogonal multiple access (NOMA) communication system. The PA is assumed to be able to be placed anywhere along the waveguide and serves two NOMA user equipment (UEs) in both uplink (UL) and downlink (DL) scenarios. Exact closed-form expressions for the average BER of each user are derived under practical imperfect successive interference cancellation (SIC). These expressions are then used to optimize the PA location for minimizing the overall average BER of both UEs. In the UL case, the interference between the users' channels introduces phase-dependent fluctuations in the BER cost function, making it highly non-convex with many local extrema. To address this challenge, a smoothing technique is applied to extract the lower envelope of the BER function, effectively suppressing ripples and enabling a reliable identification of the global minimum. In the DL case, a joint optimization of the PA location and NOMA power allocation coefficients is proposed to minimize the average BER. Simulation results verify the accuracy of the analytical derivations and the effectiveness of the proposed optimization methods. Notably, the UL results demonstrate that an optimally positioned PA can create the required received power difference between two equally powered UEs for reliable power-domain NOMA decoding under imperfect SIC.