I don’t think so. What Peter is saying is at that time David is not ascended into the heavens. The Old Testament saints are being treated differently at this time.
Act 2:25 For David speaketh concerning him, I foresaw the Lord always before my face, for he is on my right hand, that I should not be moved:
Act 2:29 Men [and] brethren, let me freely speak unto you of the patriarch David, that he is both dead and buried, and his sepulcher is with us unto this day.
Act 2:31 He seeing this before spake of the resurrection of Christ, that his soul was not left in hell, neither his flesh did see corruption.
Act 2:34 For David is not ascended into the heavens: but he saith himself, The LORD said unto my Lord, Sit thou on my right hand,
Act 2:35 Until I make thy foes thy footstool.
As you can see Peter is referring to the physical resurrection of the body, no where the soul of the believer is.