3. The desired $m$ is 595.

Let $\mu$ be the set of all two-element subsets of $S$. A two-element set $V$ is called a "good set" if and only if there exists $f \in F$ such that $f(V)$ is a single element. Construct a directed graph $G$ as follows: the vertex set is $\mu$; for $U_{1}, U_{2} \in \mu$, if there exists $f \in F$ such that $U_{2}=f\left(U_{1}\right)$, then draw a directed edge from $U_{1}$ to $U_{2}$, called an $f$-edge from $U_{1}$ to $U_{2}$. Thus, "$F$ has property $P(k)$" is equivalent to "for any $U \in \mu$, there exists a directed path of length at most $k-1$ whose starting point is $U$ and whose endpoint is a good set."

First, we prove: If $F$ has property $P(2019)$, then $F$ has property $P(595)$.

Assume $F$ has property $P(2019)$. Then for any $U \in \mu$, there exists a directed path from $U$ to a good set. Consider the shortest path with this property:
$$
U=U_{0} \rightarrow U_{1} \rightarrow \cdots \rightarrow U_{m}.
$$

Then $U_{0}, U_{1}, \cdots, U_{m}$ are pairwise distinct two-element sets.
Thus, $m+1 \leqslant \mathrm{C}_{35}^{2}=595$, which means there exists a path from $U$ to a good set with length at most 594. This proves that $F$ has property $P(595)$.

Next, construct a set $F$ such that it has property $P(595)$ but not property $P(594)$. Thus, $F$ has property $P(2019)$, and for any positive integer $m \leqslant 594$, $F$ does not have property $P(m)$.

Assume $S=\{1,2, \cdots, 35\}$, and for a two-element subset $U=\{x, y\}$ of $S$, define
$$
d(U)=\min \{|x-y|, 35-|x-y|\}.
$$

For a directed edge $U \rightarrow V$, if
$$
d(V)-d(U)=0,1,-1,
$$

then the edge is called a distance-preserving, distance-increasing, or distance-decreasing edge, respectively.
Define $f_{1} \backslash f_{2}: S \rightarrow S$ as
$$
\begin{array}{l}
f_{1}(x)=\left\{\begin{array}{ll}
x+1, \quad 1 \leqslant x \leqslant 34 ; \\
1, \quad x=35,
\end{array}\right. \\
f_{2}(x)=\left\{\begin{array}{ll}
x, & 1 \leqslant x \leqslant 34 ; \\
1, & x=35 .
\end{array}\right.
\end{array}
$$

It is easy to verify that $F=\left\{f_{1}, f_{2}\right\}$ has the following properties.
(1) $\{1,35\}$ is the only good set of $F$.
(2) All $f_{1}$ edges are distance-preserving. There are 33 $f_{2}$ edges that are not loops, among which 16 are distance-decreasing edges
$$
\{i, 35\} \rightarrow\{1, i\}(2 \leqslant i \leqslant 17),
$$
16 are distance-increasing edges
$$
\{i, 35\} \rightarrow\{1, i\}(19 \leqslant i \leqslant 34),
$$
and 1 is a distance-preserving edge
$$
\{18,35\} \rightarrow\{18,1\}.
$$

Consider the shortest path $L:\{1,18\}=U_{0} \rightarrow U_{1} \rightarrow \cdots \rightarrow U_{m}=\{1,35\}$ from $\{1,18\}$ to the good set $\{1,35\}$.

Notice that, $d\left(U_{0}\right)=17, d\left(U_{m}\right)=1$, the distance decreases by 16.

By property (2), $L$ includes each distance-decreasing $f_{2}$ edge exactly once, and $L$ does not contain any distance-increasing or distance-preserving $f_{2}$ edges (since the distance-preserving $f_{2}$ edge ends at $U_{0}$). Additionally, all edges in $L$ are $f_{1}$ edges. Thus,
$$
\begin{array}{l}
L: U_{0} \xrightarrow{\text { 34 } f_{1} \text { edges }}\{17,35\} \xrightarrow{f_{2}}\{1,17\} \\
\xrightarrow{34 \text { } f_{1} \text { edges }}\{16,35\} \xrightarrow{f_{2}}\{1,16\} \rightarrow \cdots \\
\rightarrow\{1,2\} \xrightarrow{\text { 34 } f_{1} \text { edges }}\{1,35\}.
\end{array}
$$

Thus, the total length of $L$ is $35 \times 16+34=594$, which indicates that the shortest distance from $U_{0}$ to a good set is 594.
Notice that this path traverses every member of $\mu$.
Therefore, $F$ has property $P(595)$ but not property $P(594)$.
3. Let $S$ be a 35-element set, and $F$ be a set consisting of some mappings from $S$ to itself. For a positive integer $k$, we say that $F$ has property $P(k)$: if for any $x, y \in S$, there exist $k$ mappings $f_{1}, f_{2}, \cdots, f_{k}$ in $F$ (which can be the same), such that
$$
\begin{array}{l}
f_{k}\left(\cdots\left(f_{2}\left(f_{1}(x)\right)\right) \cdots\right) \\
\quad=f_{k}\left(\cdots\left(f_{2}\left(f_{1}(y)\right)\right) \cdots\right) .
\end{array}
$$

Find the smallest positive integer $m$, such that all $F$ with property $P(2019)$ also have property $P(m)$.