Generally when "last" followed by a noun then the definite article is used. For example: "It was the last time I won." But in the news I came across this: "I can't lose and say 'I won a last time.'" why "the" is not used in the last example?
The last time refers to the most recent occasion in a series of occasions, and thus the final one, to date.
The last time I went to the beach, there was a strong rip-current.
When I saw him for the last time, he was committed to the idea of moving to Alaska.
A last time refers to such a final time as one instance of many such last times.
There will be a last time to visit a friend who is ill, a last time to kiss a lover goodbye, a last time to play Fetch with your dog, a last time to see the sunset or the sunrise. Life will have many last times...and first-times.