No, the breath of life was breathed into Adam for physical life. Apart from the physical life, the spirit lives on. You are only focusing on physical life.
And the breath of life is FROM God. God removes the breath of life from us at death. Job 34:14-15 and Eccl 12:7 specifically states this action of God. Notice the word "withdrew" in Job.
Gen 2:7 NIV Then the LORD God formed a man from the dust of the ground and breathed into his nostrils the breath of life, and the man became a living being.
Job 34:14-15 NIV If it were his intention and he withdrew his spirit and breath, 15all humanity would perish together and mankind would return to the dust.
Eccl 12:7 NIV and the dust returns to the ground it came from, and the spirit returns to God who gave it.