The secretion of osmolytes into a lumen and thereby caused osmotic water inflow can drive fluid flows in organs without a mechanical pump. Such fluids include saliva, sweat, pancreatic juice and bile. The effects of elevated fluid pressure and the associated mechanical limitations of organ function remain largely unknown since fluid pressure is difficult to measure inside tiny secretory channels in vivo. We consider the pressure profile of the coupled osmolyte-flow problem in a secretory channel with a closed tip and an open outlet. Importantly, the entire lateral boundary acts as a dynamic fluid source, the strength of which self-organizes through feedback from the emergent pressure solution itself. We derive analytical solutions and compare them to numerical simulations of the problem in three-dimensional space. The theoretical results reveal a phase boundary in a four-dimensional parameter space separating the commonly considered regime with steady flow all along the channel, here termed “wet-tip” regime, from a “dry-tip” regime suffering ceased flow downstream from the closed tip. We propose a relation between the predicted phase boundary and the onset of cholestasis, a pathological liver condition with reduced bile outflow. The phase boundary also sets an intrinsic length scale for the channel which could act as a length sensor during organ growth.