-y is a common baby-talk suffix appended to words such as pig (to make piggy), kitten (to make kitty), etc. As babe used to be a common word for infants and young children, I thought it would be likely that baby is a hypocoristic form of babe, originating from baby-talk. However, I have no way to back up this hypothesis, and am wondering if someone else knows how to prove or disprove it.
According to the Oxford English Dictionary, the word baby is formed from babe + the -y suffix. Further, they add that baby is likely from baby-talk:
The English word was borrowed into many other European languages; compare e.g. French baby (1704 in sense ‘small child’, 1793 as a term of endearment, 1898 in sense ‘doll’), Italian baby (1877), Dutch baby (1875), German Baby (second half of the 19th cent.), Danish baby (late 19th cent.). C
Compare also French bébé (c1755 as a term of endearment, 1858 in sense ‘small child’, 1885 in sense ‘doll’), which has often been considered a loan < English, but perhaps (in spite of its late attestation) shows a parallel formation < the (reduplicated) syllable /be/ , characteristic of early infantile vocalization.
Baby is attested to circa 1400, while babe is noted from the late 1300s. Note also that babe, though the origins are unknown, is likely to have formed from baby-talk as well:
Origin uncertain. Probably either < a (reduplicated) syllable /ba/ which is characteristic of early infantile vocalization (and which probably also directly or indirectly underlies baba n.7, baba n.2, baba n.5, baba n.6, babble v.1, etc.), or perhaps shortened < baban n. (which is only attested in a few texts, although its scanty attestation may be due to the fact that it belongs to an informal register, and so would have been more frequent in spoken than in written usage; with the shortening process involved, compare e.g. Gib gib n.1, Tom Tom n.1, Will Will n.3, and similar pet forms of forenames).
Both words emerged around the same time (in written usage, at least), and may both be from forms of baby-talk. If babe is from early infantile vocalization, as is noted, it is possible that baby is not actually a hypocoristic form because they are both from baby-talk. However, if babe is a shortening (the other possibility), then baby would be a hypocorism.
The OED says yes: babe + the -y suffix used to form pet names and familiar diminutives.