Natan Slifkin wrote a book "The Camel, the Hare, and the Hyrax" where he studied the statements in the Torah, the Talmud and other Jewish sources. He gives the following summary on his site:
The Torah lists four animals that possess only one kosher sign.
The Talmud, following its own principles of drawing additional meaning from words in the Torah, infers that the Torah’s list is exhaustive.
Elsewhere, the Talmud states that this topic argues for the Divine origins of the Torah, but the meaning of this is disputed:
Approach A: The simple reading of Rashi is that the argument refers to Moshe being familiar with the physiology of the four animals in the list.
Approach B: Alternately, one can argue that it refers to Moshe knowing all the local animals that possess one kosher sign.
Approach C: Tosafos explains that it refers to Moshe knowing about an animal called the shesuah, but this is a difficult explanation, as the simple reading of the verse does not indicate that the shesuah is a type of animal.
Approach D: Beginning in the eighteenth century, it was claimed that the Talmud’s argument refers to the Torah saying that there are no other such animals in the entire world. This argument rests upon (a) the boldness of the claim and (b) the veracity of it (as per point 2 above).
Making an argument from the boldness of the claim is fundamentally flawed, as there is no claim in the Torah that there are only four animals in the world possessing one kosher sign. Simply speaking, they are presented merely as examples from the region of the Land of Israel that were a particular dietary risk for the Jewish People. The idea that the list is specified as being exhaustive would only be accepted by someone with an a priori belief in the divine origins of the Talmud.
The lamoids and peccaries from South America also possess only one kosher sign. To posit that they are of the same min as camels and pigs (respectively) can only be done with a novel definition of min that grants a high degree of unspecified flexibility in categorizing new species under the Torah’s preexisting range of types. Accordingly, making an argument out of the exclusivity of the list is greatly weakened.
There is overwhelming evidence (discussed in chapters six and seven) that the shafan and arneves are the hyrax and the hare, and there are no alternative viable candidates. Positing the existence of extinct and unknown species is not viable in this case.
According to all evidence, the hare does not bring up the cud. To resolve this problem, we must say that the term ma’aleh gerah is an idiom that refers to such phenomena as ruminant-style chewing or cecotrophy, and/or to invoke the concept that "the Torah speaks as in the language of men." These approaches are viable, albeit somewhat difficult.
There are conflicting reports as to whether the hyrax regurgitates its food. Based on my own extensive observations of hyraxes, it appears that they do sometimes regurgitate small amounts. It is likely that the hyrax practices merycism, which can be defined as ma’aleh gerah without difficulty. If it does not practice merycism, then it can only be defined as ma’aleh gerah on the basis of its manner of chewing, probably requiring us to invoke the concept that “the Torah speaks as in the language of men.” As with the hare, these approaches are viable, albeit somewhat difficult.
Since we are forced to define characteristics such as merycism, ruminant-style chewing or cecotrophy as ma’aleh gerah, then there are still further types of animals that possess only one kosher sign, even with our novel flexible definition of min/type.
These further examples of animals with one kosher sign raise a problem with the Talmud, which apparently claims that the Torah’s list is exhaustive. However, there are two approaches which explain the Talmud in a way that avoids this problem:
• The Talmud is only making a statement about the exclusivity of the camel (due to it being the only ma’aleh gerah animal that is domesticated, or that lacks upper teeth, or that is a true ruminant); but the hare and hyrax may indeed share their characteristics with other animals. This only leaves the problem of the lamoids, which can perhaps be rated as a type of camel, albeit with some difficulty.
• The Talmud is only giving a rule for the general region surrounding the Land of Israel, but there may indeed be other such animals in remote regions of the world. This fits well with how various other seemingly universal rules in the Talmud are explained by other authorities - including Chazal themselves! - as only referring to local animals.
The book discusses other animals that fall under such wider classification.
The animals practicing cecotrophy: some types of rats, pikas, lemurs, coypu / nutrias, coalas (during one stage of their development), ringtail possums, capybaras, some of which obviously can be rejected as examples of sheretz.
The animals practicing merycism: cangaroos, koalas.
Among the additional animals with split hooves, which don't chew cud, the book also rejects the case of hippopotamuses, and discusses tapirs, and babirusas and concludes that despite developed stomachs these animals do not chew cud.
While Slifkin's answer reflects the present day state of science and observation of animal world, studies in genetics show that the creation of animal kinds is rather fluid (Jeanson, "Replacing Darwin"). Therefore, the rabbis in the Talmud could have been quoting the tradition from the Noach's Ark. For estimates, regarding the number of species in the Ark, one can read the articles by Jean Lightner and her collegues (Determining Ark Kinds, Mammalian Ark kinds).
