Can I use the word "puddin" to call my best friend even we are both girls?

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Nada, I imagine that by puddin you mean "a buddy". No, it's a term that is used only for men, it means a very good close friend.
Ps: I am not a native English speaker.

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Do you mean 'pudding' without a 'g'? As a term of endearment?

I'm not sure where you found this word, but 'puddin' or (even pudding) is not a normal thing to call someone, regardless of their gender. The fact that you are girls doesn't make much difference. Addressing someone as 'pudding' suggests that they are fat, so it isn't a very nice thing to call someone. There may be some couples who have particular terms of endearment, and some may call each other 'pudding'.... or pumpkin, or cabbage...or anything else. But that doesn't mean that you should copy them!

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Nada, I imagine that by puddin you mean "a buddy". No, it's a term that is used only for men, it means a very good close friend.
Ps: I am not a native English speaker.

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....and on the subject of terms of endearment, take a look at this:
http://www.bbc.co.uk/news/magazine-22699938

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