Where can I get a cordless telephone where the base station works even when there is no power (it is battery operated) ? It should have answering system and caller id.

I would personally recommend using a Cyberpower backup.  Avoid lost calls or missed messages during a power failure! Plug
your cordless phone or answering machine into CyberPower's battery
back-up unit for more than 4 hours of emergency back-up power. The
PhoneSaver provides safety, convenience and peace of mind.
The CyberPower PhoneSaver provides built in surge protection to
prolong the life of your cordless phone and answering machine. The
PhoneSaver's 4AH battery provides more than four hours of battery
back-up time. Included adapters allow the unit to work with any 6V,
7.5V, 9V or 12V cordless phone. The built-in transformer plug
replaces your phone's standard, bulky wall adapter. When a power
failure occurs, the emergency utility light activates making sure
you're not left in the dark. There's also a lifetime warranty with
a $10,000 Connected Equipment Guarantee. The PhoneSaver provides
safety, convenience and peace of mind at a time you need it most.
* Use your cordless phone or answering machine in the event of a power
outage!
* Protection from phone line surges and spikes - a leading cause of phone
degradation and failure.
* Works with all 6V, 7.5V, & 9V cordless phones and answering machines.
* Provides more than 4 hours of battery backup time.
* Emergency utility light activates automatically during a power outage.
* Built-in transformer replaces bulky standard adapter.
* $10,000 Connected Equipment Guarantee.

Hello, we just recently bought the AT&T DECT 6.0 Black/Silver Digital Corded/Cordless Answering System (CL84109) for my grandmother and I installed it for her. I can attest that this unit does have a backup battery in its base station (I believe it was two AAs but it may have been AAA i can't recall for sure) and it works when there is no power/the station is unplugged. There is a readout on the basestation which also states when the phone is unplugged, in use, etc. Overall I thought the phone was very easy to use and good quality. Its the first cordless phone in years my grandmother recieved as a gift and didn't return or give back!

The Panasonic(R) KX-FG6550 2-Line 5.8GHz Digital is a complete communications center designed just forthat purpose. It combines a 2-line 5.8GHz digital FHSS cordless phone,15-minute digital answering and a plain paper fax/copier all into 1stylish and feature-packed system. It includes a fax/copier base unitand 1 handset with its own separate charger cradle. Plus the systemsupports up to 3 more handsets for a total of 4 to support your increasing needs.  The unit also has a battery backup.  You can purchase the KX-FG6550 model online at Office Depot, Inc at a cost of $148.99.