The given name "Barbara" in English is "Barbora" in Lithuanian.
The surname is more difficult. There are these names in the current online phone book:
1. C^asas, pronounced CHA-sas (unmarried women = C^asaite; married women = C^asiene)
2. C^as^as, pronounced CHA-Shas (unmarried women = C^as^aite; married woman = C^as^iene)
3. Paseckis, pronounced Pa-SETS-kis (unmarried women = Paseckyte, sometime in older times, Paseckute; married women = Paseckiene)
4. Pas^iskas, pronounced Pa-SHIS-kas (unmarried women = Pas^iskaite; married women Pas^iskiene)
There are no names that begin with the letters "Casiesk-", "Pasiesk-", "Casesk-" or "Pasesk-".
A name ending in "-ute" is always an unmarried female where the masculine ending is "-us" or in olden times, "-is". So Barbora's father and brothers would be Casieskus/Casieskis or Pasieskus/Pasieskis.
Many, if not most, Lithuanians use cell phones, and the online directory is only for corded phones, so it is possible that there are other surname that might fit your gg-grandmother's maiden name.