I have been sifting through, trying to find a first person voice for Icarus.
John Banville writes The Sea as first person, but as a recollection of something that happened some time before. He is in the present and recalls the past and move from present to past tense accordingly.
Graham Swift writes The Light of Day in first person present tense, but again slips into past tense for recollections.
Engleby is first person past tense, but then this makes sense when at the end the first person narrator explains that the account is his diary. Sebastian Faulks cleverly keeps this secret so that the account feels more like the action as it happens.
In Moon Tiger, Penelope Lively moves between first person present tense and third person past tense as she switches between the narrative of the story teller and flash back.
Graham Green, in The Quiet American, has his narrator write about his friend Pyle in the first person past tense. You don't get any real sense of it being an account of something that happened previously; there is no nagging imperative to imagine just where and when the narrator is recounting his tale.