The state, condition, or rank of being a baby; babyhood or the characteristics associated with infancy.
From baby or babe plus -ship (a state, condition, or office). The -ship suffix was productive in English for creating abstract nouns describing conditions or ranks.
Words ending in -ship (like friendship, leadership, kingship) are all about describing states of being, so 'babeship' follows an old English pattern—though it's rarely used today because we prefer saying 'babyhood' instead.
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