In familiar French, lâcher quelqu’un can mean "to let down", and, as some sort of metaphor, with objects, "to stop working (for someone’s use)". In your example, it means, "I can’t use my hand at all, my hand won’t move".

Some similar uses:

Un voisin m’avait promis de m’aider, mais entretemps il m’a lâché (= he let me down).

Ma voiture m’a lâchée (= my car broke down, esp. just when I needed it)

Tu veux bien me lâcher ? (= leave me alone, stop harassing me)

It can also be used in a negative turn, and then it means "to harass, to chase someone"

Je ne vais pas te lâcher jusqu’à ce que tu aies fini tes devoirs (= I will be on your back until you have done your homework).

2 familiar phrases:

Lâche-moi les baskets (=leave me alone)

Je vais pas lâcher l’affaire (=I won’t give up on this, I won’t quit)

Note that un lâcheur is some sort of quitter, someone who lets their friends down. It can be used to gently nag someone.

Alors, tu viens pas avec nous, espèce de lâcheur ?