I'd use a bigger one - baby wash cloths may be small enough that she can get the whole thing in her mouth. Just wet it and wring it out good before you freeze it - you don't want it sopping wet. A good sturdy cloth, no frays.
I don't recall any problem with frost bite. :
: With anything that she's putting in her mouth you have to keep an eye on her - but I'm sure you know that already.
I don't recall any problem with frost bite. :

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