Building is raising our rent by 5% even though units in the same building are going for 20% less

Our building was raising our rent in Fremont by 8% and was lowered to a 5% raise after sending an email saying we'd like to stay but the increase is too high and other buildings in the neighborhood have cheaper units.

We replied noting that there's currently a unit in our building with equivalent square footage going for 20% less than what we'd be paying. They replied:

The current homes that are available are competing with what is currently available out in the market. When there is a saturation of apartments in the market it will drive down new lease rents. Unfortunately, this does not have an impact of renewal rates on current occupied homes.  

Should I keep pushing for a lower rate? Do we still have any leeway? We could easily leave and go to a cheaper building, but want to avoid the move (which is probably why they know they have the leverage to raise the rent even after the market has gone down).