Only cordless phones that don't have batteries in the base station. You simply need to buy a better cordless phone, or a UPS for your existing phone.
Power failure isn't so traumatic with a little advance planning. When we lose power my desktop computers and DVRs are on UPS, the cordless phones have batteries in the base station and I have emergency lighting that goes on instantly.
FWIW, landlines provide their own power so having a direct connect handset will always work in a blackout. What some people don't realize is if you live too far from down the road, away from the phone hub, or out in the country, that the phone company has a battery attached to your line that charges all time and if you use the phone too much even your landline will drop dead. I never knew this until back in the 80s a guy I knew that lived on the edge of town decided to run a BBS and his line kept dropping dead after people were on his BBS for many hours at a time.