I have never heard of that origin, and the etymology I found online does not support that pussy is derived from pusillanimous. Do you have any references for that?
http://www.etymonline.com/index.php?term=pussy
http://www.etymonline.com/index.php?term=pusillanimous&allowed_in_frame=0
Try Webster’s Collegiate Dictionary, in editions before about 1970, for the etymology that created the word “pusillanimous”. Most sources online have “streamlined” their references, and a lot of associations that were commonplace a generation ago have fallen into disuse or abandoned altogether.
“Pussy” did used to mean “lacking in courage”, synonymous with “wimp”. Might be where we get the hybrid word “wussie”.