The first rabbit to go into space did not have a name. It went into orbit with a Samoyed dog named Otvazhnaya. They went into orbit from Russia on July 2, 1959. This information is from a book entitled 'The Top Ten of Everything' published by Dorling Kindsley.
Jun 27 2000, 6:15 PM
zbeckabee
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The rabbit was named Marfusha (Martha).
Otvazhnaya (Brave One), Snezhinka (Snowflake), and their rabbit partner Marfusha (Martha) made a high altitude test flight on July 2, 1959 aboard an R2-A rocket. All three animals were recovered successfully. Otvazhnaya made five high altitude test flights from 1959 to 1960.
http://www.silverdalen.se/stamps/dogs/library/library_space_dogs_russian.htm