The generalisation is not true, for various reasons.
Firstly, if $l=2$ then just as in ordinary arithmetic there will usually be two solutions for the $l$th root. For example, $1$ and $-1$ are both square roots of $1$ modulo any $z$; and if $z>2$ then $1\not\equiv-1$.
Secondly, in many cases the square root may not exist, again just as in ordinary arithmetic. For example, modulo $3$ the complete list of squares is
$$0^2,\ 1^2,\ 2^2\equiv0,\ 1,\ 1,$$
and so $2$ does not have a square root.
Moreover, odd order roots (which always exist and are unique in real arithmetic) may not exist or may have multiples values in modular arithmetic. For example, the cubes modulo $7$ are
$$0^3,\ 1^3,\ldots,\ 6^3\equiv0,\ 1,\ 1,\ 6,\ 1,\ 6,\ 6,$$
so $0$ has a unique cube root, $1$ and $6$ have three each, and $2,3,4,5$ have none.
Another point worth considering: in real and complex arithmetic it is true that any number has at most $l$ roots of order $l$. Even this is not true in modular arithmetic: for example, $1$ has four square roots modulo $15$.