I don't know if she's still around somewhere but I have located the marriage of Andrea Baiada to Stefana Gallina the 19th of January 1862 Carini. From there I take her back to my ancestor (we are related) Vincenzo Baiada who married 20 Jan 1737 Vincenza Rubbino. It easily traced from there to Michele Baiata or Baiada who married 1571 Filippa Lo Cilip. This Michele was a native of Carini and born about 1545. His parents were most likely natives as well. The Malta theory is most likely just family folklore.
Here is the direct male line:
Andrea Baiada (son of Vincenzo Baiada & Anna Jiacopello) married Stefana Gallina 19 Jan 1862 Carini
Vincenzo Baiada (son of Giacomo Baiada & Maria Mazzola) married Anna Jiacopello 27 Jul 1817 Carini
Giacomo Baiada (my ancestor and the son of Vincenzo Baiada & Vincenza Rubbino) married Maria Mazzola 1782 Carini
Vincenzo Baiada (son of Michele Baiada & Sabbina Maranzano) married Vincenza Rubbino 20 Jan 1737 Carini
Michele Baiada (son of Giacomo Antonino Baiada & Gerolama Campo) married Sabbina Maranzano 8 Sep 1692 Carini
Giacomo Antonino Baiada (son of Michele Baiada & Lauria di Vita) married Gerolama Campo 8 April 1663 Carini
Michele Baiada (son of Giacomo Baiada & Vincenza Cajetta) married Lauria di Vita 15 June 1636 Carini
Giacomo Baiada born 26 May 1573 Carini (son of Michele Baiada & Filippa Lo Cilio) married 20 April 1599 Vincenza Cajetta
Michele Baiata or Baiada, native of Carini, born about 1545 married 1571 Filippa Lo Cilio, she was a native of Trapani, Sicily
That about sums it up, the records before 1570 are being extracted at this moment I may be able to take the line further back.
I hope she finds this!
All records cited can be found on the original resources pages of my web sitewww.cariniexchange.com
If ever I can help anyone with Carini, please email me!