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Let F be a field with nontrivial non-Archimedean valuation of rank one and let X be a zero-dimensional Hausdorff space. The vector space C(X, F) of all continuous functions from X into F is provided with the compact-open topology c. We prove that C(X, F, c) is bornological if and only if X is a Z-replete space.doi:10.1090/s0002-9939-1980-0548100-7 fatcat:x3t466i3lbftnb2fmunzjuzvli