Mouse-deer do chew cud (they has four stomachs), but have not two, but four hoofed toes.
Are they kosher?
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According to Bikkores HaTalmud (Vienna, 1863, pp. 387-9), by Rabbi Yaakov Yechezkel HaLevi (cited by R Natan Slifkin here, page 10, and footnote 28), the "java mouse deer" is the shafan, which is classified as a non-kosher animal (Devarim 14:7). The same source posits that the "greater mouse deer" may be the arneves, which is also forbidden in the same verse.
However, R Slifkin finds this opinion to be problematic for a number of reasons, see op cit., ibid; see also the other articles cited by R Slifkin in that footnote there (28).
Thank you @GershonGold for posting this link ;-)
According to the various kashrus books that I have seen, Rav Hirsch, Rabbi Art Scroll, and others, a "split hoof" is a single covering of the bottom of the foot split completely in half (from front to back). Thus, the fact that the picture of the mouse deer shows four toes and not a single completely split hoof would mean that it is not kosher.