An innovative way to produce quantum Hall ribbons in a cold atomic system is to use M hyperfine states of atoms in a one-dimensional optical lattice to mimic an additional synthetic dimension.″ A notable aspect here is that the SU(M) symmetric interaction between atoms manifests as infinite ranged″ along the synthetic dimension. We study the many-body physics of fermions with SU(M) symmetric attractive interactions in this system using a combination of analytical field theoretic and numerical density-matrix renormalization-group methods. We uncover the rich ground-state phase diagram of the system, including unconventional phases such as squished baryon fluids, shedding light on many-body physics in low dimensions. Remarkably, changing the parameters entails interesting crossovers and transition; e.g., we show that increasing the magnetic field (that produces the Hall effect) converts a ferrometallic″ state at low fields to a squished baryon superfluid″ (with algebraic pairing correlations) at high fields. We also show that this system provides a unique opportunity to study quantum phase separation in a multiflavor ultracold fermionic system.