Volumetric bioprinting (VBP) is a light‐based 3D printing platform, which recently prompted a paradigm shift for additive manufacturing (AM) techniques considering its capability to enable the fabrication of complex cell‐laden geometries in tens of seconds with high spatiotemporal control and pattern accuracy. A flexible allyl‐modified gelatin (gelAGE)‐based photoclick resin is developed in this study to fabricate matrices with exceptionally soft polymer networks (0.2–1.0 kPa). The gelAGE‐based resin formulations are designed to exploit the fast thiol‐ene crosslinking in combination with a four‐arm thiolated polyethylene glycol (PEG4SH) in the presence of a photoinitiator. The flexibility of the gelAGE biomaterial platform allows one to tailor its concentration spanning from 2.75% to 6% and to vary the allyl to thiol ratio without hampering the photocrosslinking efficiency. The thiol‐ene crosslinking enables the production of viable cell‐material constructs with a high throughput in tens of seconds. The suitability of the gelAGE‐based resins is demonstrated by adipogenic differentiation of adipose‐derived stromal cells (ASC) after VBP and by the printing of more fragile adipocytes as a proof‐of‐concept. Taken together, this study introduces a soft photoclick resin which paves the way for volumetric printing applications toward soft tissue engineering.