Polymers are widely used in daily life, but exhibit low strength and low thermal conductivity as compared to most structural materials. In this work, we develop crystalline polymer nanofibers that exhibit a superb combination of ultra-high strength (11 GPa) and thermal conductivity, exceeding any existing soft materials. Specifically, we demonstrate unique low-dimensionality phonon physics for thermal transport in the nanofibers by measuring their thermal conductivity in a broad temperature range from 20 to 320 K, where the thermal conductivity increases with increasing temperature following an unusual ~T1 trend below 100 K and eventually peaks around 130-150 K reaching a metal-like value of 90 W m-1 K-1, and then decays as 1/T. The polymer nanofibers are purely electrically insulating and bio-compatible. Combined with their remarkable lightweight-thermal-mechanical concurrent functionality, unique applications in electronics and biology emerge.