Atmel ATtiny EEPROM partially erased on power cycle
Such a slow decrease allows many possibilities for the MCU to "go nuts" and corrupt the EEPROM. It has enough supply voltage to do bad things (like write to EEPROM), but not enough supply voltage to operate reliably. 

The same thing could happen with a relatively fast decrease, just less often, so you may be lucky to have caught it before it caused field failures. 

A BOR circuit is the answer, hold it in reset whenever the supply voltage is out of spec for the clock speed and chip that you're using. 

I don't recall if that particular chip has an on-board BOR and if it does, whether it works well enough, but external reset chips such as ADM805/809 variants work well provided you have the reset pin available.