So far I have only found usage examples which show the difference in amount of two things by two or more degree, for example
She earns five times as much as I do.
Smokers are 10 times more likely to develop lung cancer than nonsmokers, two times more likely to develop heart disease.
I wonder if it is possible to say that X is ONE TIME more likely to do Y.
You would never in normal speech say "This is one times that". You wouldn't say "This bag is one times as big as that bag". You would say "These two bags have the same size.
So you would never say "This is one times as likely as that". You would say "These two are equally likely".
Notice your dictionary definition states that "times" can be "used to show the difference in amount", and not to show that the two amounts are equal. On the other hand it is correct to say "This is twice as likely as that". Using "twice" is more idiomatic than "two times" in many situations.
(Of course in elementary mathematics it is correct to say "one times five equals five".)
I suspect you've misunderstood what "5 times more likely" or "5 times as much" means. It means multiply. If one person has £500 and another has £100, the first person has 5 times more money than the second, not 4 times more.
So logically "X is one time more likely to do Y" would just mean "X is equally likely to do Y", and it would be a strange and convoluted way of saying it and would probably confuse your readers.
