Relay with contact resistance less than 10mohm
You may be able to use a MOSFET for your design.
Depending on your design, you may be able to use a MOSFET with a very low RDSon (resistor when  the FET is on)

There are some MOSFETS that have lower ON resistance than 10mohm.

I could have searched for a MOSFET for you but I don't have your schematics or specifications.

You should be able to find a good match at http://www.vishay.com/ (There are more MOSFET manufacturers)