It might be possible to deposit a metallic film on a sheet of plastic. It is however, a problem with respect to the bonding of that layer. Depending on the plastic used, it might not have a strong bond with the metal, which would then over time develop defects.

On the other hand, it might be possible, when carefully manufacturing this, that the plastic has 'functional groups' (parts of the molecule with a specific function) that would bind a metal. Note however, that in this case, the functional groups might also react with other chemicals. Almost no functional group will only under go one reaction, especially not when metals are involved. Therefore, this would not be the most durable of options. Furthermore, because of the interface between, most likely, an amorphous solid (wikipedia, amorphous solids) and a metal structure is bound to create some defects.

A last option I can think of, is the use of conductive polymers. These polymers, because in many ways they mimic metals, also have a similar 'shininess' to them. However, I'm not sure about their really being a mirror. 

Hope this helped.

Can flexible plastics have reflective coating (vacuum metalized)?
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