James Anderson was dismissed on the penultimate ball of the match at Headingley on Tuesday as England, on the verge of an improbable draw, lost by 100 runs to concede a home series to the Lions for the first time.
The result has seen Cook's future as skipper, already in a precarious state following the embarrassing 5-0 Ashes whitewash in the winter, questioned again, but the 29-year-old has insisted that he will not quit and has vowed to turn England's fortunes around.
"It's a huge honour to captain England and I'm in it for the long haul, as long as I can be, because I believe I'm the right man for the job," he told Sky Sports News.
"If someone else decides that not the case then so be it, but I've got to give all blood, sweat and tears into the England captaincy. I never quit on anything, I'm a determined bloke and I'm determined to turn this around."
To add to the left-handed opening batsman's mounting worries, he has not made a Test match century in over a year and has scored just 601 runs at an average of 25 in his last 24 innings.