Let $p\geq 3$ be a prime number and let $k\geq 1$ be some integer.

Is it always true that if $x^2\equiv 1\pmod{p^k}$ then $x\equiv\pm1\pmod{p^k}$ ? 

For $k=1$ it is true since $x^2-1\in\mathbb{F}\_p[x]$ is a degree $2$ polynomial with two distinct roots $-1,1$ and so any other root in $\mathbb{F}\_p$ must be one of them. 

I know that there exists a primitive root modulo $p^k$.