Are film capacitors affected by DC bias?
Film capacitors have a very stable capacitance over DC bias. Look at this:



But it comes to happen that film capacitors are not the only ones immune to DC bias:

 

So, the actual question is why do ceramic capacitor have such a bad behaviour with DC bias? And the answer lies in the dielectric.

In order to achieve high capacitance values in small capacitor sizes, a high-K dielectric materials like Barium Titanate have to be used. Unfortunately Barium Titanate also has ferroelectric properties that reduce its capacitance value under DC bias:


  When spontaneous polarization is reversed under no voltage stress (no
  DC bias), MLCCs achieve a high capacitance. However, if an external
  bias is applied to the spontaneous polarization process, the free
  reversal of spontaneous polarization is much more difficult. As a
  result, the capacitance gained is lower compared to the capacitance
  before the application of the bias. This is why capacitance decreases
  when DC bias is applied-hence the term DC bias characteristic.


Sources:

Quote: Understanding DC Bias Characteristics in High-Capacitance MLCCs.

Images: this comparison document from NIC Components.