The baby "deer-mouse" became just the 43rd living member of this species in Europe when it was born on April 9 in the Fuengirola Biopark near Malaga.
Originating in southeastern Asia, the deer is so called because its tiny dimensions and big eyes make it look more like a rodent, despite its tiny hooves.
At birth the baby -- which has not yet been named because it is still too small to determine its sex -- weighed about 100 grams (nearly four ounces).
But "it is growing very fast", a spokeswoman for the nature park, Asun Portillo, told AFP on Friday.
The deer-mouse typically grows to about the size of a rabbit and weighs about a kilo (about two pounds) when fully grown.
"It is doing very well, in its enclosure, although it cannot suckle yet and cannot feed by itself."
Its mother has lived in Fuengirola since 2007 and its father was brought over from Lille, France a year ago, the park said.
The survival of the species, known by scientists as "tragulus javanicus", is threatened by deforestation in its native southeast Asia, the park said.