No, but here is a workaround.  On the first page of the form, put the
formfields at the top of the page, but not in the header.  Set up your
document so that it either has a different first page header in section 1 or
has a continuous section break after the formfields you just added with a
blank header for section 1.  In the header for additional pages, use REF
fields.  For example, you have a formfield at top of first page which is
named "Text1", in header of additional pages you have a field { REF Text1 }.
In the options for the formfields, you check 'Calculate on exit', then the
REF fields automatically update when you exit the formfields they reference.

Regards,
Chad

It might be possible, but you couldn't use it because the header is locked
by document protection. What is it that you are trying to accomplish? Is the
information you want in the header going to be anywhere else in your
document?