Aims. We use stellar line-of-sight velocities of Antlia B, a faint dwarf galaxy in the NGC 3109 association, to derive constraints on the fundamental properties of scalar field dark matter originally proposed to solve the small-scale problems faced by cold dark matter models. Methods. We use the first spectroscopic observations of Antlia B, a distant (d $\sim$ 1.35 Mpc) faint dwarf ($M_\text{V} = -9.7$, M$_\star \sim 8\times10^5$M$_\odot$), from MUSE-Faint - a survey of ultra-faint dwarfs with the Multi Unit Spectroscopic Explorer. Through measurement of line-of-sight velocities for stars in the $1'\times 1'$ field-of-view, we identify 127 stars as members of Antlia B, allowing us to model its dark matter density profile with the Jeans modelling code GravSphere. We implement a model for scalar field dark matter into GravSphere and use this to place constraints on the self-coupling strength of this model. Results. We find a virial mass of ${M_{200} \approx 1.66^{+2.51}_{-0.92}\times 10^9}$ M$_\odot$ and a concentration parameter of ${c_{200}\approx 17.38^{+6.06}_{-4.20}}$ for Antlia B. These results are consistent with the mass-concentration relations in the literature. We constrain the characteristic length scale of the repulsive self-interaction $R_{\text{TF}}$ of the scalar field dark matter model to $R_{\text{TF}} \lesssim 180$ pc (68% confidence level), which translates to a self-coupling strength of $\frac{g}{m^2c^4}\lesssim 5.2 \times 10^{-20}$ eV$^{-1}$cm$^3$. The constraint on the characteristic length scale of the repulsive self-interaction is inconsistent with the value required to match the observations of cores of dwarf galaxies in the Local Group, suggesting that the cored density profiles of those galaxies are not caused by scalar field dark matter.