[Kryptid (OP)]

Virtual particle-antiparticle pairs continually pop into and out of existence in the vacuum. Would the space between some such pairs have expanded so rapidly during inflation that they would have been prevented from annihilating and returning to the vacuum? Would inflation have therefore resulted in a flood of new particles and antiparticles popping into existence?

[Colin2B]

In an intense em field it could. If you look at http://physics.princeton.edu/~mcdonald/e144/e144trans_052897.pdf down to “Strong Field Pair Creation as Barrier Penetration” there are some details of separation/field strength. I think there was an experiment with laser that produced some pairs.
Also in intense spacetime curvature pairs can get separated eg Hawking Radiation.

[Bill S]

That link is too technical for me, but one question that arises is:  Which field is referred to in the following?

For a virtual e +e − pair to materialize in a field E the electron and positron must separate by distance d sufficient to extract energy 2mc2 from the field:

eEd = 2mc2 . 

The answer is probably there, somewhere, but it eludes me; unless, of course, it's the electron field, but that would raise other problems for me. 

[Colin2B]

Which field is referred to

It’s the electric field. Unfortunately you don’t get something for nothing, you have to put in energy in form of the electric field.

[Bill S]

It’s the electric field.

And the electric field is already there, because, as Matt Strassler says: "A field is something that is present everywhere in space and time"?

[Colin2B]

And the electric field is already there,

But if it’s value is zero or less than that required by the formula you won’t get pair production.