Hello Ruben,
No. I had a BEFSR81 Router for a period of years and it never altered my outside IP address.
You could try resetting it. From the BEFSR81 Manual:
The Reset button can be used in one of two ways:
1. If the Router is having problems connecting to the Internet, press the Reset
button for just a moment with a paper clip or a pencil tip. This clears up any
jammed connections, and is similar to pressing the Reset button on your PC
to reboot it. This will reboot the Router while keeping all of the Routers settings.
2. If you are experiencing extreme problems with the Router and have tried all
other troubleshooting measures, press the Reset Button and hold it down
until the red Diag LED on the front panel turns on and off completely.
This will restore factory defaults and clear all of the Routers settings,
including settings such as IP addresses or a new password.
You could also try getting firmware for your router. The latest firmware I had was V2.45.10
If these things fail, you might consider that a BEFSR81 router is well past it in terms of age and you might consider a new router. A Cisco Linksys RV042 is an excellent and modestly-priced router. I have one of these now at home. It is well capable of supporting a static connection and my dynamic connection changes so rarely I can access it from elsewhere. And then, if it does not work (maintain a static IP), then it is your ISP's problem.
... Thinkpads_User
