The answer is no. E.g., assume that $n=2$, $\xi_i=X_i$, and $X_1,X_2,\nu$ are any random variables (r.v's) each with values in the set $\{1,2\}$ and with $P(\nu=1)=p\in(0,1)$. With these conditions in place, the dependence between $X_1,X_2,\nu$ may be arbitrary. For instance, we may suppose that $\nu$ is independent of $X_1,X_2$, or we may suppose that $\nu=X_1$, or ... .
Take any $c\notin\{1,2\}$ and let $Y_1,Y_2$ be obtained from $X_1,X_2$ by replacing $X_\nu$ by $c$, so that
\begin{equation}
(Y_1,Y_2)=\left\{
\begin{aligned}
(c,X_2) &\text{ if }\nu=1,\\
(X_1,c) &\text{ if }\nu=2.
\end{aligned}
\right.
\end{equation}
Then
$$P(Y_1=c,Y_2=c)=0\ne p(1-p)=P(\nu=1)P(\nu=2)=P(Y_1=c)P(Y_2=c).
$$
So, $Y_1,Y_2$ are not independent.