I did a new clean installation of Windows XP with SP2.
Downloaded Mozilla Firefox 43.0.4
I can download the Flash Player from here:
File size is 18.1 Mb.
However, when I try to install the Flash Player the installer does not run when selected from downloads or when double clicked on from within the downloads folder.
Would installing SP3 solve this problem?
It's possible this is a certificate issue. XP SP2 does not support the latest cryptographic technologies necessary to sign binary installers. SP3 does have this support. XP is no longer supported by Microsoft and I don't know if you'll be able to find an SP3 installation.
I installed SP3 for Windows XP.
Then I downloaded the Flash Player installation file for version 188.8.131.526 and it now runs and has installed normally and is working fine.
Updating and installing SP3 after performing a new clean install of XP with SP2 was the solution for me.
Good news. Since SP3 supports the latest certificate signing technologies, you shouldn't encounter this issue again.