The Exarchate of Ravenna fell long before the empire did. However, it did last a full 167 years and its borders may be less unstable than they'd appear.
If you examine the territorial extent of the Exarchate of Ravenna in the 590s, it's actually pretty similar to the area of Roman colonization in 100 BC, except more extensive.
So there is precedent for a territory ostensibly smaller than the Exarchate of Ravenna defeating a territory larger than the Lombard states. Discounting of course, the decline in the wealth of Rome.
Exarchate of Ravenna in 590 AD:
Rome, colonies, and "allies" in 100 BC:
Furthermore, most of the important parts of the Exarchate of Ravenna survived to 650 AD, when the rest of the empire was really starting to feel the effects of falling apart:
Eastern Empire in 650 AD:
With that in mind, could it be possible, with proper investment, reinforcement, and leadership from the East, for the Exarch of Ravenna to reconquer the rest of Italy or at least stabilize his position until the Caliphate is formed?
In that case, if Anatolia and Constantinople fall to the Umayyads, could the Exarchate of Ravenna become the main surviving piece of the Empire, bringing the Romans back to where they began?