Not as such. Headers are a section property.
However, you can certainly have replicating text.
Labels in Word are
Tables . All of the tools available for tables are available to work on labels and all of the limitations of tables apply to labels. Tables can have replicating rows known as headers but they cannot reproduce within the table, they only show up when the
table breaks across a page.
See
Repeating Data (Populating Multiple Like Fields).
You can horizontally split the rows that make up the labels to make them your "headers."
