Because Krishna appears to be just like an ordinary human being, people often mistake Him as such and erroneously think His body to be material, just like ours. However, this is not the case, as will be scientifically explained in this nice article by Bhaktin Reena Menon.
Yes. Lord Krishna is the adi-purusha, the sat chit ananda swarupa.. Eternal, blissful and full of knowledge. He is super conscious spirit. Omnipresent and omnipotent.
Our material bodies are made of 5 gross elements and are subject to birth, death, old age and disease. We are neither omnipotent nor omniscient. We, as part and parcel of Parabrahman Krishna possess similar divine bodies as sat chit ananda swarup in our original state devoid of material body and qualities. However we belong to His marginal potency with an inherent tendency to fall in material consciousness.
Lord Krishna on the other hand is achyuta Or infallible. He is eternally free from material contamination being the Supreme Brahman. Lord Krishna's divine leelas and killing of ferocious demons right from His divine appearance is a proof of His eternal divine body.
Foolish persons or people who are conditioned in material consciousness speculate the Lord's activities and conclude them to be that of a mortal. Lord Krishna in Bhagavad Gita says that His appearance, disappearance and activities are all divine and those who conclude it as birth, death and activities based on karmic bondage are certainly in illusion.
Lord Krishna reveals Himself only to His unalloyed devotees. No amount of austerity, severe penance, charity or good karma can equate pure devotional service in the heart of a bhakta. Pure devotion unto His lotus feet is the only shelter and way to attain salvation.