I am by no means an expert but I think this is indeed possible, just look at the Huronian glaciation for example.
This ice age led to a mass extinction on Earth.
Of course only microorganisms have lived back then (and mostly died because of oxygen, not necessarily the cold). But since ice isn't the best soil for plants a huge portion of the flora would die and if there are no plants then herbivores would also starve, which then leads to a starvation of carnivores as well which rely on those herbivores (and omnivores as well for both reasons). Probably only marine life and animals like penguins, seals or polar bears would survive because sea life could still thrive in this environment (like they do even today near the poles) as long as there is liquid water.
Whatever causes such an extreme icing would probably also lead into a global ice age, especially since ice has a very high albedo so the incoming sunlight gets reflected a lot better than from water or rock. And if less sunlight gets absorbed the earth gets colder which then accelerates the process.
At at least one point in earth's history even the entire earth was covered in ice, the Marinoan glaciation. I'd assume that this would definitely lead into a mass extinction for basically every living organism. (Despite not mentioning a mass extinction in the article, it is listed as an extinction event).
Even if the ice only covers let's say Australia and southern African areas like Madagascar and only kills those plants and animals (which would most probably happen since they aren't made for such cold environments), this would cause a huge loss in diversity, since the majority of plants and animals only live there and nowhere else on the planet.
98% of its land mammals, 92% of its reptiles, 68% of its plants and 41% of its breeding bird species exist nowhere else on Earth
(...) is home to one of the most diverse and unique floras in the world, with over 210 vascular plant families, and 50-80% of species being unique to the state in the largest of these families.
As someone in the comments of OP's question pointed out, it would also largely depend on the cooling rate but given the fragility of the ecosystem in Madagascar I don't think this would be a huge factor. Especially since they can't flee from the island. Probably some species would try to escape to the African mainland once the ice has created a bridge but at least plants, cold blooded animals and small birds wouldn't make it. Same with Australia.
Hope this satisfies your question.