In this paper, we investigate a fluid antenna system (FAS)-assisted downlink non-orthogonal multiple access (NOMA) for short-packet communications. The base station (BS) adopts a single fixed antenna, while both the central user (CU) and the cell-edge user (CEU) are equipped with a FAS. Each FAS comprises $N$ flexible positions (also known as ports), linked to $N$ arbitrarily correlated Rayleigh fading channels. We derive expressions for the average block error rate (BLER) of the FAS-assisted NOMA system and provide asymptotic BLER expressions. We determine that the diversity order for CU and CEU is $N$, indicating that the system performance can be considerably improved by increasing $N$. Simulation results validate the great performance of FAS.