Non-invertible mapping of a random vector
Solution 1:
No. To consider a simple counterexample, assume $X\sim \mathcal{N}(0,I_N)$, the standard normal distribution and $f(x_1,...,x_n)=1\{x_1\geq 0\}$. Then $Y$ is binary and thus the support of $Z$ has at most two values.