Not by definition, by experience. Interestingly enough, and in spite of this common experience, there is no law of conservation of volume. The total volume of the universe may be both finite and increasing. Keep in mind in what follows that the common experience does not include such things as the constancy of light's speed in all inertial frames, or of the physics that occur at extremes of temperature.
Perhaps you are correct and the volume of the universe is infinite and the finite mass/energy is expanding within it. That is not the big bang theory. And accumulating evidence makes the infinite volume theory harder to sustain.
The big bang is something of a misnomer because it is not an explosion like a bomb within a preexisting space exploding with shrapnel flying in all directions as you seem to suggest.
The big bang is assumed to encompass then entirety of the finite universe at all times. The superhot pre-atomic early phase was the state of the ENTIRE universe. Cooling and development of the various forms of matter occurred simultaneously with an increase in space. Thus there is a relationship between mass, energy, and space such that all points in the universe share the same perpective.
How can space increase? How can the perspective be the same everywhere and the universe still be finite? How can there be no "edge"?
Part of the answer to these questions comes from the theory of general relativity (a necessary presupposition of the big bang theory) in which there is an equivalence between a "static" observer in a gravitational field and one that is accelerating "through" space. An equivalence. An equivalence between space and "no space" that depends upon gravity which in turn depends upon matter--matter that is increasing as the universe cools.
The current big bang theory (which of course is always under revision as new evidence presents itself) holds that all existence (a.k.a. the universe) is finite in volume, mass, and energy, is increasing in volume, is the same everywhere (w.r.t. observing its expansion), has a finite history and yet has always existed (since time started with the big bang).
I'm no cosmologist. Maybe one lurking out there can correct me, and also give a clearer explanation of what is known about the mass/energy/volume/time interrelationships.
One final note of the metaphysical elegance of the finitude suggested by the big bang theory. Infinity is not a closed concept. Infinity makes conceptual sense only as a process, like time or volume growing toward infinity--yet time and volume are always still finite. In short, the application of infinity to extent or amount (rather than to an open-ended process) is a violation of the principle of identity.
Thus observations that have led to the big bang theory have, without intending to do so, seem to have preserved one of the most fundamental axioms of thought--the principle of identity.
Who has made such a claim for "conservation of volume"? Not I.
the constancy of light's speed in all inertial frames
While initial acceptance has been slow, there are compelling arguments for variable speed of light (VSL) in the physics community. VSL is not a widely accepted view, but breakthroughs are often vehemently rejected at the outset. The jury is still out, at minimum.
Perhaps you are correct and the volume of the universe is infinite
I didn't say our universe was infinite, or even that it will grow indefinitely (a weaker condition). In fact, I suggested that ours might not be the only universe. The void might encompass multiple universes. It is also possible that the separation between these universes is so great that no one can be observed from another (i.e. the light-years of separation between universal envelopes is much greater than the age of the universes).
The big bang is assumed to encompass then entirety of the finite universe at all times.
Our universe, I suppose. It can't encompass everything at any given time though, can it, or there wouldn't be anything to expand with respect to. Even then, the fact of our universe, even if adequately described by the "big bang" (assumed, not proven, as you note), does not rule out multiple big bangs, nor multiple universes. It does not even rule out multiple big bangs from the "speck" from which ours theoretically emanated.
On a final note, I must take exception to "the metaphysical elegance of the finitude suggested by the big bang theory". In fact, it leaves at least two metaphysical holes big enough to drive a galaxy through--the issue of cause, and the issue of demarcation, i.e. what lies beyond the finitude of our universe. There is an elegant solution to those, and it remains forever beyond human physics.