You're looking for a material that has plastic deformation properties and is highly ductile. To achieve your desired material behavior, you'll want a polymer like Parafilm, which you can buy online.
edit: More generally, you should look at different thermoplastics and find one that meets your needs. Available material properties at room temperature range from extreme malleability/ductility (“floppy” plastic like Parafilm) to relative stiffness and structural integrity (you need to heat these materials, although this can be easily achieved with readily available hot water in examples like Ingo).