Can someone be saved if they willingly live in sin?

Then when I'm dying in my bed, I'll just ask him to forgive me for my sins.

This is an attempt to logically undermine the point of Salvation but it suffers the logical flaw that the person saying this is already accepting the logic of Salvation in an attempt to disprove Salvation.

The standard answer to this bit of illogic is the question:

And if you are walking across the street and not paying attention and get hit by a bus and killed instantly - how are you going to do that, exactly? The morgues are full of people who didn't expect to be unexpectedly dead. Pray that you aren't one of them. Except that since you renounced prayer until your deathbed, you can't do that.

Generally that gets the commenter muttering about lets change the conversation because you have demolished their logic trap.

I would like to let the Bible answer this question directly.

In the Gospel of John chapter 8, Jesus provides what I believe is a wonderfully simple and straight forward answer to this kind of question.

You will remember that the Pharisees brought a woman to Jesus who had been caught living a life of sin... I would like you to refer particularly to Verse 11 (i add the bold highlighting)

My take on the example below is that, in knowing our sins are forgiven and that we are no longer condemned, Jesus makes it quite clear what is expected of us...we must choose from that point forward to leave the former life of sin.

1 but Jesus went to the Mount of Olives.

2 At dawn he appeared again in the temple courts, where all the people gathered around him, and he sat down to teach them. 3The teachers of the law and the Pharisees brought in a woman caught in adultery. They made her stand before the group 4 and said to Jesus, “Teacher, this woman was caught in the act of adultery. 5 In the Law Moses commanded us to stone such women. Now what do you say?” 6They were using this question as a trap, in order to have a basis for accusing him.

But Jesus bent down and started to write on the ground with his finger. 7 When they kept on questioning him, he straightened up and said to them, “Let any one of you who is without sin be the first to throw a stone at her.” 8 Again he stooped down and wrote on the ground.

9 At this, those who heard began to go away one at a time, the older ones first, until only Jesus was left, with the woman still standing there. 10 Jesus straightened up and asked her, “Woman, where are they? Has no one condemned you?”

11 “No one, sir,” she said.

“Then neither do I condemn you,” Jesus declared. “Go now and leave your life of sin.”

Can someone be saved if they willingly live in sin? Since God forgives sins, why not live a sinful life? Then when I'm dying in my bed, I'll just ask him to forgive me for my sins.

Can such a one be saved? The answer is yes. Whoever calls upon the name of the Lord shall be saved and there is no scriptural evidence to suggest that one must clean up their act prior to calling upon the Lord. The question here actually becomes will that one, having intentionally and selfishly contradicted God lifelong, suddenly renounce self and cry out for mercy on the deathbed? Scripture paints this as both unlikely and foolish for, one the one hand:

There is a hardening process and a "giving over" principle outlined in Scripture for our warning:

As it is said, “Today, if you hear his voice, do not harden your hearts as in the rebellion.” For who were those who heard and yet rebelled? Was it not all those who left Egypt led by Moses? And with whom was he provoked for forty years? Was it not with those who sinned, whose bodies fell in the wilderness? And to whom did he swear that they would not enter his rest, but to those who were disobedient? So we see that they were unable to enter because of unbelief. - Hebrews 3:15-19

And on the other hand there is reliance upon a future day of opportunity that has not been guaranteed:

Come now, you who say, “Today or tomorrow we will go into such and such a town and spend a year there and trade and make a profit”— yet you do not know what tomorrow will bring. What is your life? For you are a mist that appears for a little time and then vanishes. Instead you ought to say, “If the Lord wills, we will live and do this or that.” As it is, you boast in your arrogance. All such boasting is evil. - James 4:13-16

Today, right now, is the only opportunity to call on the name of the Lord:

“In a favorable time I listened to you, and in a day of salvation I have helped you.” Behold, now is the favorable time; behold, now is the day of salvation. - 2 Corinthians 6:2

However, I suspect the question might actually be asking, "Is such a one as this saved?". That is to say, in accordance with Jesus' declaration in John 3 that one must be born again in order to see and enter the kingdom of God, "Has someone who willingly lives in sin lifelong been born again?" The answer is no:

See what kind of love the Father has given to us, that we should be called children of God; and so we are. The reason why the world does not know us is that it did not know him. Beloved, we are God's children now, and what we will be has not yet appeared; but we know that when he appears we shall be like him, because we shall see him as he is. And everyone who thus hopes in him purifies himself as he is pure. Everyone who makes a practice of sinning also practices lawlessness; sin is lawlessness. You know that he appeared in order to take away sins, and in him there is no sin. No one who abides in him keeps on sinning; no one who keeps on sinning has either seen him or known him. Little children, let no one deceive you. Whoever practices righteousness is righteous, as he is righteous. Whoever makes a practice of sinning is of the devil, for the devil has been sinning from the beginning. The reason the Son of God appeared was to destroy the works of the devil. No one born of God makes a practice of sinning, for God's seed abides in him; and he cannot keep on sinning, because he has been born of God. By this it is evident who are the children of God, and who are the children of the devil: whoever does not practice righteousness is not of God, nor is the one who does not love his brother. - 1 John 3:1-10 

This is not to say that one who is born again is sinless because then we would not have need of an advocate with the Father who is faithful to forgive us of and cleanse us from our sins when we confess them:

If we say we have no sin, we deceive ourselves, and the truth is not in us. If we confess our sins, he is faithful and just to forgive us our sins and to cleanse us from all unrighteousness. If we say we have not sinned, we make him a liar, and his word is not in us. - 1 John 1:8-10

The saved person has become a new creature (2 Corinthians 5:17), is implanted with God's seed and born of Him (1 John 3:9), is indwelt by the Spirit of truth (John 14:15-17), and has entered into a process which was entirely foreign prior to the new birth:

But I say, walk by the Spirit, and you will not gratify the desires of the flesh.  For the desires of the flesh are against the Spirit, and the desires of the Spirit are against the flesh, for these are opposed to each other, to keep you from doing the things you want to do. But if you are led by the Spirit, you are not under the law.  Now the works of the flesh are evident: sexual immorality, impurity, sensuality,  idolatry, sorcery, enmity, strife, jealousy, fits of anger, rivalries, dissensions, divisions, envy, drunkenness, orgies, and things like these. I warn you, as I warned you before, that those who do such things will not inherit the kingdom of God. But the fruit of the Spirit is love, joy, peace, patience, kindness, goodness, faithfulness,  gentleness, self-control; against such things there is no law. And those who belong to Christ Jesus have crucified the flesh with its passions and desires. If we live by the Spirit, let us also keep in step with the Spirit. - Galatians 5:16-25

The Spirit of God within warring against the natural desires! This is particular exclusively to those who are saved and is a real sign of life (Romans 7).

In Summary

Saved (born again) people will still commit sin but it will not be a dedicated lifestyle over the long haul because the Spirit of God within them actively opposes it. The sanctification of the Spirit (1 Peter 1:2) is a process begun at the new birth and it is unto obedience: It is a work of God. The ability to contentedly refuse God and knowingly sin, banking on forgiveness at the 11th hour (as proposed in the question) is clear indication that the process has yet to begin. Such a person can be saved but there is no guarantee that they will for if God's voice is heard clearly enough to be actively refused in favor of sin then repentance and forgiveness are not merely being put off for another day, they are being spurned. This is wickedness and "the lamp of the wicked will be put out" (Proverbs 24:20)