Carmat shares were down by 12.4 percent at 29.55 euros in mid-session trading, wiping some 26 million euros ($28 million) off the company's stock market value.
Carmat Chief Executive Stephane Piat said in a statement on Wednesday that the artificial heart was not involved in the patient's death, and that the company's last three implantations had functioned normally.
A spokeswoman for the French drugs agency said it had requested further information from Carmat following the patient's death. She did not say when implants could restart.
The 76-year old who died had Carmat's artificial heart implanted at the end of August and is its fifth patient to die.
He was the first of 20 patients in a second so-called 'pivotal' phase of tests, which is a prerequisite to market the product in the European Union.
The first death of a patient with one of Carmat's devices fitted, occurred in 2014, less than 80 days after his operation. He was aged 76.
Carmat shares are down by around 30 percent since the start of 2016.
"As a reminder, it took two years to get the greenlight to start human implantation and two and a half more years to validate the feasibility study on only 4 patients," Romain Zana, an analyst with Exane BNP Paribas, told clients in a note.
"The mid-2017 target set to finish the pivotal trial looks, accordingly, completely unrealistic and the likelihood of bringing the device on the market is further declining," he added.
Carmat has yet to generate any significant revenue but its products could represent a major medical breakthrough if proven reliable as heart diseases represent a leading cause of death worldwide.
In 2014, Carmat had said around 100,000 patients in the United States and Europe could benefit from its artificial heart, a market worth more than 16 billion euros.
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They are designed for people with end-stage heart failure, a deadly condition where the heart is unable to pump blood adequately around the body, and therefore are an alternative to heart transplants given the shortage of donor organs.
Among Carmat's competitors are privately-held SynCardia Systems and Abiomed, both from the United States.
SynCardia's artificial heart is the only one approved both in the United States, Canada and the European Union, and has been implanted in more than 1,625 patients as they wait for a heart from a matching donor. The longest a patient has lived with the device is just under four years prior to a transplant.
Carmat's heart is designed to serve not as a bridge to transplant but as a permanent implant, extending life for terminally ill patients who cannot hope for a real organ, generally because they are too old and donors too scarce.