That's a very interesting question, that cannot be answered by a simple yes or no.
You see, "infinite" is not a unit of measure, it's a rate of change. What's more, infinity (or at least, Cantor's theorems on infinity) have variations. The most common "infinity" that most people refer to are is really Cantor's "zeroth" infinity (sometimes refered to as null-infinity, null-aleph or 1-dimensional infinity)...
I wrote a little paper on infinities and Cantor's infinities a while ago... I may still have it somewhere if you're interested. To oversimplify, iofftopicgine a 1-dimensional line; infinetly long. We both can agree that the line goes on for as long as you measure it.
Now, take a finite piece of that line, of any finite length you want. Put a marker halfway on that line. Now, put a marker halfway on both halves of that line. Repeat... infinitely... you will have an infinite number of points on the line - so we can agree that a finite line can have an infinite number of points. This is Cantor's zeroth infinity.
Now, lets go back to our infinite line. You can have an infinite number of finite lines, each with it's own set of infinite points. This is Cantor's first infinity.
Next, take two infinite lines, and make a 2D grid out of the two (X/Y for now). You can have an infinite number of points, on an infinite grid, on two infinite axises. This is cantor's second infinity.
The third goes three dimensional, the fourth stretches my understanding...
Now, going back to your universe.... If you use Cantor's notion of infinity being a rate of change, not a unit of measurement; yes, the universe is infinite. You can travel in a straight line forever without stopping. Problem is, you can do the exact same thing in a mobius strip... (or a football for that matter)...
So we then come to the notion of navigating in a straight line, forever, without ever passing over our previous path... is it infinite in this sense? Well... we can only theoreticise...
One of the more interesting notions is that of the inverted ball... Iofftopicgine being an ant on the inside of a football. Your "universe" is infinite in that you can walk straight forever without stopping... but you will pass "over" yourself, as well as the "volume" or "area" of your universe being finite. Now, invert you ball, (a proceedure that is quite complicated as you can never really "fold" any sides - this is a whole research of its own accord). Your "exploded" ball/universe is now infinite in volume, infinite in mass, and infinite in distance... problem is, you need to have something to "expand" your universe into.
So we get to the big bang theory - where two point of infinitesimally small mass, but infinitesimally large energy (unstoppable object vs. unmovable one?) collided and the energy converted to matter... but into what?
This is where theories mix with religion, and where theories are wilder than comics... I try to stay out of there, but I do keep a keen ear and like hearing the latest fantasy-oops , I meant theory
M.