And it’s also possible for a species to fail to be a clade. Imagine, for instance, a species that has a thriving and stable population on one island, but can’t fly or swim (or at least, not enough to leave the island). Along comes a storm, and carries a few of them on a debris raft to another island. The rafters breed and set up a new population on the new island, but conditions there are just different enough from the original island that, in adapting to it, they diverge into a new species. In such a situation, the new species would be a proper clade, but any clade that included the old species would also have to include the new one.
According to modern taxonomic criteria, a species essentially has to be a clade, or rather, has to include all members of a clade (from whatever branching point defines the origin of that clade). In the example you cite, the population on the original island, and the population on the new one, both become separate clades when the new one branches off from the original one. But if the population on the first island was established from colonists from the mainland, they both form a single clade relative to the mainland population. You therefore have to treat the populations on the two islands the same relative to the mainland population. You can’t treat the population on one island as the same species on the mainland, and the other one as distinct. Either both are subspecies of the mainland species, or both are separate species.
So what happens when sea levels drop, land bridges form between all three, and the mainlanders and Island 1 start interbreeding but Island 2 stays aloof from both of them?
For that matter, the concept of a clade itself breaks down as soon as you can have multiple lines of ancestry. For instance, some modern H. sapiens have some Neanderthal DNA, meaning that there must have been at least some interbreeding between the two populations (even if it was rare). So it’s possible (even if not now, then back in the time before Neanderthals died out) to have two mostly-sapiens individuals whose most recent common ancestor was a Neanderthal.
Basically, when we designate a population as a species, we are stating a hypothesis, that this lineage will continue to remain distinct over evolutionary time. This is indicated by such terms as Evolutionary Species Concept and Evolutionarily Significant Unit. In the case of allopatric populations, we normally can’t test such a hypothesis experimentally.
In the case you propose, the hypothesis would be refuted in the case of island 1, but upheld for island 2. That on island 1 would be reclassified to be a member of the same species as that on the mainland, but that on 2 would be maintained.
While it is theoretically possible that a population like that on island 2 could acquire reproductive isolating mechanisms when that on island 1 didn’t, off hand I can’t think of any real-world cases. (This would be in the case of isolated populations on the mainland with a similar kind of relationship that came into contact after isolation.)
Here in Panama, the Escudo Hummingbird occurs on the tiny island of Escudo de Veraguas off the Caribbean coast of Panama. It is 50% larger than its mainland relative, the Rufous-tailed Hummingbird. In my opinion, it might well not interbreed with the mainland form were they to come into contact. However, genetic work has shown that it does not form a separate clade from the mainland forms, so it is currently classified as a subspecies of it.
It is important to realize that species status is a property that belongs to a population, not an individual. A first-generation hybrid does not technically belong to either parent species. Although species are sometimes described on the basis of a single specimen, that is because we believe it is representative of a larger population. Taxonomists these days try to avoid describing species based on single specimens; in the past many such species have turned out to be aberrant individuals or hybrids.
It is also important to understand that in molecular systematics clades are not (or shouldn’t be) defined on the basis of single genes. Each gene is inherited somewhat independently, and forms its own clade. When a gene has been acquired by introgresion through hybridization, it will show show a discordant pattern of descent from the rest of the genome.
These days taxonomic clades are defined on the basis of multiple genes, and subject to complex analyses in order to generate a consensus tree. For an example relevant to the case in the OP, take this reference.
Considering the case of Neanderthal genes in Homo sapiens, each gene has its own tree of descent from Neanderthals (which is how we recognize them), as does each sapiens genes. But taken as a whole, the sapiens genome is a single clade in population terms.
In the case of modern humans, your most recent common ancestor with respect to Neanderthal genes may be a Neanderthal. But you have many more recent common ancestors for your sapiens genes. And considering the entire genome, sapiens forms a single clade.
Speaking of lizards, there is also the interesting case of the New Mexico Whiptail. The New Mexico Whiptail ( Aspidoscelis neomexicanus ) is a parthenogenetic hybrid of the Little Striped Whiptail ( Aspidoscelis inornatus ) and the Western Whiptail ( Aspidoscelis tigris ). I can see how the New Mexico Whiptail could be considered a separate species under several of the different species definitions mentioned earlier in this thread because it can reproduce and maintain a stable population in the absence of the Lesser Whiptail and Western Whiptail. However, I have observed Little Striped Whiptails, Western Whiptails, and New Mexico Whiptails in the same location on the same day. Do the Lesser Whiptails and Western Whiptails continue to produce hybrids from time to time? I don’t know. If they do, would this introduction of new genetic material into the New Mexico Whiptail population affect its classification as a separate species under any of the different species definitions we have discussed so far?
There are lots of species of obligately parthenogenetic reptiles, amphibians, and fish. The Biological Species Concept can’t really be used for them or other species that reproduce asexually. Instead, something like the Evolutionary Species Concept has to be applied. I haven’t been able to find confirmation that new crosses of Lesser and Western Whiptails are occurring; I think that Lesser Whiptails originated from a hybridization event in the past.
Interestingly, most such parthenogentic species seem to be short-lived. Genetic recombination is needed in order to adapt to changing environments over the long term, so species that can’t do this tend to become extinct rather than adapt.
A totally non-relevant comment: I noticed you said Lesser Whiptail instead of Little Striped Whiptail. When I first typed my post I said Lesser Whiptail as well. Then I went to Wikipedia just to make sure I didn’t have any spelling errors. Wikipedia said Little Striped Whiptail so I thought maybe I was just stupid and changed it. I feel a little less stupid now.
I didn’t really check to see what the actual English names were.
Another useless comment: People that classify the Prairie Lizard as a different species separate from the Fence Lizard include the Sceloporus lizards in Arkansas in the Prairie Lizard species. But nobody in Arkansas would call the native Sceloporus lizards Prairie Lizards. They would call them Fence Lizards (if they knew what to call them at all).
That’s the problem with Common Names. The same species has different names in different places, and different species often have the same name.
I once wrote a term paper on those little buggers when I was taking genetics 
. Hard as hell to catch, I tell you what.
They’re actually a bit of a potential conundrum for the ESC as well. The reason being is IF you have multiple hybridization events between the same two bisexual species, which is not at all unlikely considering how common the unisexual lineages are. Because you then have multiple evolutionarily static (or mostly static, adaptation reduced to random mutations rather than recombination) gene lines that are very similar/almost identical morphologically, karyologically and ecologically. But they are on separate evolutionary trajectories, because as parthogens there is no interbreeding except for backcrossing with yet another bisexual species. And that traditionally has led to yet another species designation, like the triploid A. neotesselatus. There have even been tetraploids.
Philosophically speaking under the ESC one could argue every hybridization event produces a more or less reproductively isolated novel unisexual species. Which is…problematic, to say the least. Best to just ignore gene lines and lump them for the sake of practicality 
.
I only saw one Dark-eyed Junco in the higher elevations today. It was the pink-sided variety.