Can an EEPROM be one-time-programmable?
I did find one for At25020B, but not for one with the NB suffix. Why would Atmel make a chip and not publish the datasheet? 


Why wouldn't they?

Chips are often manufactured for specific customers with specific needs, especially if the numbers are large enough.

Probably, though, your IC is an AT25020 of some kind, which pretty much sets the way it communicates.


  I think the N in some cases mean "Leadless Chip Carrier" and in some cases "One Time Programmable", since they are not related. Do you agree?


No. The wording strongly suggest it's a leadless, one time programmable part. Not either. Both.


  Looking at the picture, it obviously has 8 leads, so it is not leadless, and thus N must mean one-time-programmable. Do you agree?


No, it seems that naming scheme simply doesn't apply to the IC you have.


  As far as I understand the EEPROM technology it is by definition erasable/rewritable. 


Wikipedia: EEPROM stands for Electrically Erasable Programmable Read-Only Memory; so yes, by definition, it's electrically erasable. That functionality is controlled by an on-chip erase controller. If you simply disable that after the first write (e.g. by automatically burning a fuse), then it becomes one-time programmable.

Remember, you do not directly interface the memory cells with the pins of your IC; you talk to some kind of controller that takes the messages ("hey, can you give me data from address 0xDEAD?") and interacts with the actual memory based on its own discretion;.


  The background for even asking is because I have not had luck with communicating with the chip. A friend, who is an experienced tingler tried to with Bus Pirate and command line tool flashrom, without luck. 


What's the chance of that IC simply being broken?