Hola Clara

In this dictation this sentence "su vida está en mis manos" is an expression. We can use possessive with noun "mano". I think you may be thinking about this other sort of sentences where we use parts of our bodies and we talk about routine actions related to higiene.

"Lávate las manos! -> Wash your hands!

Here we don't use the possessive before it is understood that you want the person to wash his/her own hands, so we use the article "the" instead.

Another example would be: "Pon la cabeza en el cojín para descansar" -> Put your head on the cushion to rest.

Hope it helps

Inma