For lore, we need to look to earlier editions of the game.
According to Polyhedron #159 (2003), the special ability of the silver sword is a powerful magical enhancement, not a property of the weapon's material. The creator must be a githyanki spellcaster of at least 12th level. Whatever special technique allows their creation is either a secret to the githyanki or a power limited to them, and all but a handful of radical separatists are unwaveringly loyal to the Lich-Queen, the ultimate ruler of their race.
The same source also warns that the githyanki have specialist hunters called sword stalkers, warrior-mages who track down stolen silver swords. They even have their own ten-level prestige class. Silver swords are only ever presented by the Lich-Queen to her high ranking knights.
Silver swords actually vary considerably in power. The most powerful +5 vorpal silver swords are described as minor artifacts, which can no longer be created, even by the Lich-Queen herself; these rare weapons pre-date her rise to power.
The adventure Seekers of the Silver Forge, appearing in Dungeon #125 (2005), is the best source of information on the exact creation process of the silver swords. It takes place in the underwater ruins of Kamyn-Dhun, a githyanki astral city which escaped to the material plane, and was once known to produce the finest silver swords.
According to this adventure, the silver swords are initially produced by githyanki smiths in massive forges, seemingly by mundane methods. The resulting weapons are silvered, though not yet infused with their famous magical properties at this stage.
Next, they are purified and prepared in a special chamber engraved with hundreds of silver runes, which contains a silver and stone sword stand. This purification ritual requires 24 hours of uninterrupted meditation. This is much easier when you're a githyanki on the astral plane and have no need to sleep.
The base weapons are then infused with their magical properties in a massive psionic foundry known as the Silver Forge. This process requires a powerful spellcaster and takes several days. The forge's source of power is psionic energy stolen from unwilling captives who are kept chained nearby.
The materials used to craft the swords are described, though not named specifically, only to say that they are of excellent quality and can be used to create magic silvered weapons. It's unspecified whether these materials are special to the astral plane or also available on the material plane. Surviving githyanki cities in the astral plane are still readily able to find this material to craft silver swords, and we know from the existence of astral driftmetal that the githyanki do mine islands of solid matter floating in the astral plane.
According to the 4th edition source Dragon #377, silver swords are hand-crafted by the githyanki Lich-Queen Vlaakith herself. This not entirely consistent with her 3rd edition statistics (Dragon #100) which show that while she can create rings and wondrous items, she cannot create magic weapons, nor does she have any skill in blacksmithing. She can use wish to create a silver sword, but the cost of doing so is prohibitively expensive compared to the Silver Forges. However, it is implied that the Vlaakith of 4th edition is a different, later Vlaakith, so her abilities may be different.