Llanelli MP Nia Griffith said "everything" will be discussed in the wake of the loss of the English safe seat.
She failed to give Mr Corbyn her full support in a BBC Radio Cymru interview.
Mr Corbyn has said he will not step down in the wake of the loss to the Conservatives.
Copeland had been held by Labour for more than 80 years - the Tories' Trudy Harrison won with 13,748 votes to Labour's Gillian Throughton's 11,601.
Speaking on Post Cyntaf, Ms Griffith said: "The Conservatives are very strong at the moment so there'll be a lot of questions raised now within the party.
"What's important now is discussing the whole thing and what happened in Copeland. Why has it happened and what is the best way forward?"
Asked if the loss raised questions about Jeremy Corbyn's leadership, she said: "There will be a lot of discussion about every aspect of our party now and that will probably form a part of it".
When asked if she would resign to trigger another leadership ballot, she added it was important the party rebuilds and needs to be better prepared for future elections.
Meanwhile Gerald Jones, a Labour shadow Welsh minister, told BBC Radio Wales the result was "a significant set back for us as a party and one that we have to now rebuild from".
The Merthyr Tydfil MP said the Labour party has to "take stock" with a "lot of work to do".
Mr Corbyn said while it was "a day of disappointment in Copeland" he would not be quitting as Labour leader.
Asked if he would fall on his sword following the defeat, he said: "No. I was elected leader of this party - I'm proud to lead this party."
Jeremy Corbyn's future will form part of discussions around the fall out of the Copeland by-election, Labour's shadow defence secretary has said.