I did my FAA PPL in the UK; it's possible.
What I think is not possible is training totally ab initio outside the USA because one is then flying wholly on the privileges of the U.S. Student Pilot Certificate which (IIRC) is valid only in the USA. I haven't checked this out though.
Most European pilots doing FAA licenses in Europe already hold a European license which entitles them to be legally PIC on the training and the checkride.