No, it doesn't.
For example: consider an infinite number of marbles in an infinitely long line, all of them are touching each other.
Now, get an infinite number of helpers stationed along the infinitely long line, and have them all move the marbles in front of them to the right and stagger them out in the process, so that there's now an inch between each marble.
You now have an infinite number of marbles an infinitely long line, but it's a longer line than it started out, because there's now an inch between all the marbles.
As it stands, there's not only the expansion, but the universe also seems to be flat -- meaning it's spatially infinite, not just borderless. That observation is also compatible with a very, very small curvature, but in that case the size would still be so large that it wouldn't be practically possible to circumnavigate even if it wasn't expanding.
Space expension just means that the metric used to measure distance increases with time. It doesn't imply a border.
Consider the function equivalent to the Euclidian distance on a line : d(x1,x2) = sqrt((x2-x1)^2)
Now, you can consider the function : d2(x1,x2,y) = y * d(x1,x2)
If you view x1 and x2 as positions on one axis and y as a position on the other one, you can think of d2 as the distance between x1 and x2 expanding as you go higher in the plane.
Space expansion works in the same way but y is actually time.
Imagine you are baking sweet rolls with raisins in them. When you are proofing your dough, the yeast makes the dough expand in each point, so all the raisins become further apart.
Now imagine starting with an infinite dough. As it is proofing, it will still be infinite, but there will be a measurable increase in the distance between raisins, and the speed of how quickly a specific raisin moves away from another one increases linearly with how far they are apart when you start measuring.
That's all "expansion of space" really is. We can see other stars moving away from our own, and the stars far away are moving away faster than those close to us. And since looking outwards is looking backwards in time, we can only see as far away as corresponds to the first stars that were formed.
And maybe you've heard that we can observe "expansion is slowing down". That is actually the fact that we measure a deviation from linearity in speed vs. distance.
* All space was contained in the big bang, so "everywhere simultaneously" makes sense from that perspective.
* ...but the idea that space is expanding implies an outer "edge" (, which doesn't make sense if space isn't infinite. (or is it infinite, and just getting "more" infinite, in the way that the space on the number line between 0 and 1 is infinite, but "less" infinite than between 0 and 2?