We report the synthesis, crystal structure and magnetic properties of a new molecular complex based on a Tb(iii) ion supported by 2-furancarboxylic molecules: {Tb(α-fur)₃(H₂O)₃}_n (α-fur = C₄H₃OCOO). Two slightly different Tb sites (A and B) exist depending on the position of one of the dangling ligands. Ab initio calculations predict that, for both sites, the magnetic ground state is highly anisotropic (g_z∗ = 17.8) and consists of a quasi-doublet with a small gap, well isolated from the next excited state. The α-fur ligand forms 1D polymeric chains of Tb ions of the same type (either A or B) running along the c-axis. The crystal structure is formed by the supramolecular stacking along the a-axis of 2D layers containing parallel chains of the same type. Static magnetization and heat capacity measurements show that, magnetically, the system can be modeled as an ensemble of Ising chains of non-Kramers Tb ions with effective spin S∗ = 1/2, antiferromagnetically (AF) coupled by a weak intrachain interaction (J∗/k_B = -0.135 K). At very low temperatures, the static susceptibility reflects the presence of a 2-4% concentration of defects in the chains. Ac susceptibility measurements at H = 0 performed down to mK temperatures have enabled us to observe the slow relaxation of magnetization through two different pathways. They are assigned to Single-Chain-Magnet (SCM) behavior in two different types of AF chains (A and B), triggered by the existence of defects breaking the chains into segments with short-range order. At temperatures below 0.1 K this mechanism is replaced by individual relaxation of the ions through direct processes. Under the application of a magnetic field the system slowly relaxes by two distinct direct processes, strongly affected by a phonon bottleneck effect.