Story about a computer that thinks it's a little girl (possible Doctor Who spoilers)

Prompted by Doctor Who - “Silence in the Library”, as discussed here .

I agree with the Dopers who feel that the little girl must be the library’s computer…but when watching this episode, I was driven mad trying to remember the short story (maybe a short-short?) that had the premise of an AI’s personality being developed by making it think it was a little girl.

IIRC, the computer held the records of servicemen killed in a war, and when they allowed the AI personality to finally access the data, the personality was swamped by the flood of information.

Anyone remember this story?

ETA: Space added to avoid mouseover spoilers: I forgot this hasn’t aired in the US yet!