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You should be able to hit the button a second time and stop the blinking. Failing that, removing the power source (= unplugging the base unit: please wait fifteen seconds before reconnect) should reset everything.
You shouldn't have to re-enter the date and time if this isn't the "master" base unit.
You shouldn't have to live with blinking.
What is the PRECISE make and model of the system?
What PRECISE button did you press?
$5 is a membership fee. If you solicit help, why should it be free? If powering down didn't solve the problem, then there's more going on than what you reported: you didn't just "press the wrong button."
"Power down" may not be complete unless you remove the batteries from the handsets.
Try powering down EVERYTHING to out-of-the-box status before throwing up your hands.
Did you also try following the REST of the instructions for recording a memo: for instance, SAY SOMETHING, then press STOP?