There are files with fillable but not savable forms. Acrobat Reader does not allow save such documents.
What free software can convert fillable forms pdf to fillable and savable forms pdf?
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This thread on the Adobe forums indicates that there is no such free software. You'd have to buy Adobe Acrobat, and use LiveCycle Designer, which is included in the bundle.
If I remember right, Okular was able to save the information of a form in a pdf.
Assuming you can print these filled forms, you could use a virtual printer to create a PDF file.
I don't know of any free ones which have worked as claimed. I found PDFill to be a great tool for working with PDFs. PDFill has some free tools, but the ability to create forms is not free. It is very inexpensive compared to alternatives. There might be a free trial. Check out their web site to see if it will work for you.