Evidently yes in some cases.
IRS Publication 503 states:
U.S. citizens and resident aliens living abroad. If you are living abroad, your care provider may not have, and may not be required to get, a U.S. taxpayer identification number (for example, an SSN or an EIN). If so, enter “LAFCP” (Living Abroad Foreign Care Provider) in the space for the care provider's taxpayer identification number.
Also, see Vanessa's answer above about the child needing an SS number or ITIN.
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