Sundew plants (Drosera) capture insects using tiny drops of a viscoelastic fluid. These mucilage droplets are typically tens of micrometers in diameter, corresponding to fluid volumes in, or below, the nanoliter range. In contrast to other carnivorous plants, the physical principles and the role of rheology in the capturing mechanism are not yet fully understood. The rather simple chemical composition reported for the capturing fluid (a high molecular weight acidic polysaccharide composed of a D-glucurono-D-mannan backbone with alternating monosaccharide sidegroups) is in stark contrast to the compositional and structural complexity of other biological materials with strong extensional responses, including fibers, silks, or mucins. Here we show that the strategy used by the sundew plant combines the effects of high fluid viscosity, extensional viscoelasticity, capillary thinning and a liquid-to-solid transition driven by solvent mass transfer and aided by extensional strain-stiffening. More generally, we demonstrate how the rheological material functions of microscopic biopolymer samples in the sub-nanoliter range can be measured using an interferometry-based microrheometer for shear deformations and a capillary thinning microrheometer for extensional deformations.