Development of fiber‐spinning technologies and materials with proper mechanical properties is highly important for the manufacturing of aligned fibrous scaffolds mimicking structure of the muscle tissues. Here, the authors report touch spinning of a thermoplastic poly(1,4‐butylene adipate)‐based polyurethane elastomer, obtained via solvent‐free polymerization. This polymer possesses a combination of important advantages such as 1) low elastic modulus in the range of a few MPa, 2) good recovery ratio and 3) resilience, 4) processability, 5) nontoxicity, 6) biocompatibility, and 7) biodegradability that makes it suitable for fabrication of structures mimicking extracellular matrix of muscle tissue. Touch spinning allows fast and precise deposition of highly aligned micro‐ and nano‐fibers without use of high voltage. C2C12 myoblasts readily align along soft polymer fibers and demonstrate high viability as well as proliferation that make proposed combination of polymer and fabrication method highly suitable for engineering skeletal muscles.