Bionic man ignoring the odds is old American (4)
I believe the answer is:
inca
'old american' is the definition.
(the Inca were a South American people)
'bionic man ignoring the odds' is the wordplay.
'ignoring the odds' indicates alternate letters (letters in odd-numbered positions are removed).
'bionic'+'man'='bionicman'
The alternate letters of 'bionicman' are 'INCA'.
'is' acts as a link.
(Other definitions for inca that I've seen before include "South American (historical)" , "Original Peruvian" , "Old Peruvian people" , "who lived in S America" , "Ancient Andean Indian" .)