Polyethyleneimine (PEI), a cationic polyelectrolyte, finds great utility as a nonviral gene transfection vector. The mechanism of gene delivery process across the cell membrane through PEI-based polyplexes is still much debated; however, a general consensus is that the proton binding-based PEI conformational changes occur as the pH alters during the process. Taking a step back, even the understanding of wide pH range (from 1 to 10) dependent conformational changes for neat PEI in explicit water remains elusive. In pursuit of this objective, using a combination of optical and electron microscopy, we observed that a dilute aqueous solution of linear or branched PEI (M-w ranging from 0.8 to 750 kDa) at room temperature and having pH in the range 2.5-4 undergoes a completely novel and highly unprecedented slow self-assembly process to form a micrometer-sized thick fibrillar network. These self-assembled structures are highly robust, irreversible, and interestingly, generic for PEI and form over 24-72 h in a dilute aqueous solution irrespective of the molecular weight and configuration of PEI. A combination of turbidity and acid titration experiments on PEI aqueous solutions having different pH values reveal that the mechanism of this hierarchical self-assembly (between pH 2.5-4) can be explained by the unique protonation behavior of PEI chains and their ability to undergo conformational/morphological transitions.