In Dublin, Ireland were denied their second win of the competition when Louis Picamoles barged his way over in the 74th minute before Frederic Michalak slotted the conversion for a 13-13 draw.
And Kidney admitted that for his players it felt like another defeat, having previously lost to England and Scotland, but he hopes it's not the end of an era just yet.
"It is a decision for Brian obviously, but I am hoping it won't be his last as he is such a huge asset," said Kidney.
"It is frustrating we did not manage to close out the deal.
"It was the type of game where chances were at a premium and it is all about taking chances. These games can be decided on tiny margins.
"The players have taken everything that's been thrown at them and just got on with it.
"So to put our foot forward like that and be creating in the way we are doing actually adds to the frustration.
"The players are down, but that's the measure of how much they push themselves and why I stand by them so much.
"In another time the team would be so much stronger for what's happening at the moment."
The draw with France came at a cost for Kidney who saw Eoin Reddan, Luke Marshall, Donnacha Ryan, O'Driscoll, Peter O'Mahony, Conor Murray and Fergus McFadden all pick up knocks to add to an already lengthy injury list.
Kidney added: "I've never experienced an injury run like this, but you work your way through and I spoke to (captain) Jamie Heaslip about it and we agreed not complain."