Disclaimer: This does not answer the original question, since I overlooked a crucial assumption. I don't think it's a valid reason for deletion - this answer probably does no harm. Please don't upvote, though ;)
As requested in the comments, I present the construction of $7$ points in a periodic configuration, with each point moving at velocity $1$.
To begin with, imagine that there is a point $A$ and a ball (my name for point, to avoid confusion) $x$ such that it is guaranteed that if $x$ reaches $A$ and then collides with another ball, then it will return to $A$ after a fixed time (say, $\Delta t =1$). Then we can construct a periodic configuration with three additional points, as follows:
Pick points $B,C$ so that the triangle $ABC$ is equilateral with edge length $1$. Suppose $x$ is about to reach $A$ at time $t=0$. At time $t=-0.5$, place balls $a,b,c$ at midpoints of $BC,CA,AB$ respectively, with $a,b$ directed towards $C$ and $c$ direcred towards $A$. At time $t = 0$ we have two collisions: $c,x$ at $A$ and $b,a$ at $C$. As assumed, at $t=1$ the ball $x$ will return to $A$. At time $t=0.5$, all $a,b,c$ are at midpoints of appropriate sides, with $a,c$ moving towards $B$, $b$ moving towards $A$, and it's $0.5$ time untill $x$ reaches $A$ --- so it's exactly the same as at $t=-0.5$, up to relabelling. Thus, the balls will keep moving like this indefinitely.
To construct the configuration for $7$ balls, just take two equilateral triangles $ABC$, $A'B'C'$ with side lengths $1$ nad $\lvert A A' \rvert = 0.5$, place a ball $x$ with velocity $1$ between $AA'$, and repeat the construction above for both $A$ and $A'$.
For instance, at $t= -0.5$ it could be the case that $x$ and $b'$ are at $A'$, $x$ moving to $A'$ and $b'$ moving to $C'$; $a'$ and $c'$ are at $B$, $a'$ moving to $B'$ and $c'$ moving to $B'$; $a,b,c$ are at midpoints of $BC,CA,AB$ respectively, with $a,b$ moving to $C$ and $c$ moving to $A$.
This construction can be used directly to provide examples for any odd number no less than $11$: just add a $2k$-element periodic cycle hinted at in the comments. Unfortunately, this does not work with $9$ balls, but I believe the above idea can be extended to work for $9$ balls as well. I think it should be possible to find a periodic configuration with $5$ balls + extra ball, and then use it to construct a periodic configuration with a tringle connected to a pentagon (much like the above was a triangle connected to another triangle).
I think the above shows that, if there is no periodic configuration with $5$ balls, then it a rather special case and a peculiarity of number $5$ - not a general property of odd numbers.