We are aware of this issue with the online installer, and are working on a fix. In the mean time, please try using the offline installer posted at the bottom of the Installation problems | Flash Player | Windows page, in the 'Still having problems' section.
I also have the same issue in W8.1 and Firefox 35.0.
Correct me if I am wrong, but laszlo.m, I think same solution applies for us all, i.e., we need to directly download the installer from the website mentioned by Maria instead of the 1MB launcher available in Adobe Flash Player Install for all versions, which is the one failing.
You just need to select the browser as the links are 'smart' to check your OS and they directly download the one that suits your needs (Windows x86, x64, MacOS etc..).
I have done so and could install it without problems.