The controlled texturing of surfaces at the micro- and nanoscales is a powerful method for tailoring how materials interact with liquids, electromagnetic waves, or biological tissues. The increasing scientific and technological interest in advanced fibers and fabrics has triggered a strong motivation for leveraging the use of textures on fiber surfaces. Thus far however, fiber-processing techniques have exhibited an inherent limitation due to the smoothing out of surface textures by polymer reflow, restricting achievable feature sizes. In this article, a theoretical framework is established from which a strategy is developed to reduce the surface tension of the textured polymer, thus drastically slowing down thermal reflow. With this approach the fabrication of potentially kilometers-long polymer fibers with controlled hierarchical surface textures of unprecedented complexity and with feature sizes down to a few hundreds of nanometers is demonstrated, two orders of magnitude below current configurations. Using such fibers as molds, 3D microchannels are also fabricated with textured inner surfaces within soft polymers such as poly(dimethylsiloxane), at dimensions and a degree of simplicity impossible to reach with current techniques. This strategy for the texturing of high curvature surfaces opens novel opportunities in bioengineering, regenerative scaffolds, microfluidics, and smart textiles.