Warren Gatland's men were on the wrong end of a Dan Carter masterclass in Dunedin and could be further hampered for the second Test in Hamilton after Stephen Jones was sent for tests on an injured thumb.
And with Wales yet to win a Test in the Land of the Long White Cloud, physio Mark Davies revealed the squad would be without Bishop and possibly missing Jones too.
"Andrew has a spiral fracture to the third metacarpal bone in his right hand," said Davies.
"The injury is not complex but he is in plaster and will be out of action for the next four weeks."
"Stephen has a sore thumb. He has been X-rayed but the initial results are inconclusive and we are arranging for a further scan to fully determine the extent of the injury.
Scarlets centre Jonathan Davies is likely to take over from Bishop in the middle while Dan Biggar replaced Jones in Dunedin.