Cats do strange things. It's as much a fact of life as the sun rising in the east and setting in the west. They fight shadows, they sleep in bathroom sinks, they run into and right back out of a room completely unprompted. But their balancing skills are truly unmatched, right?
Despite cats' abilities to learn their names yet wholeheartedly ignore you when you talk to them, daintily jump to the floor no matter their perch and fiercely defend themselves against dust motes, it doesn't make them invincible. But we're always talking about how cats have nine lives, though, aren't we? I had a cat who disappeared into the woods for a month and came back looking like she'd just been on vacation -- there's gotta be something in their DNA that keeps them keeping on.
The nine lives thing is mostly steeped in longtime wonder at how cats do the things they do. Their agility and dexterity has always surprised people, along with the fact that they're always landing on their feet no matter how they fall. All this has to do with is their righting reflex, though, or a cat's body's ability to balance out and right itself while in midair.
It's an old belief; the proverb, "A cat has nine lives. For three he plays, for three he strays and for the last three he stays" is pretty ancient; even Shakespeare referred to it in "Romeo and Juliet." If we're looking for concrete evidence, though (what little there is), we can turn to ancient cultures like the Egyptians, who of course considered cats sacred and believed they were gods.
A few different things come into play here. For example, Atum-Ra, the Egyptian sun god, was thought to take the form of a cat on jaunts to the Underworld. According to legend, the god birthed eight other gods -- resulting in nine godly lives.
The specific number of cat lives can also come from a number of cultures' (Chinese and Greek, for example) belief in the Trinity of all Trinities; the number 9 represents symbolic entities in a number of traditions and religions.
So your feline friend may be able to pull off impressive balancing acts and can potentially see ghosts that we can't, but they're just as mortal as any other animal!