  We present a construction of a measure-zero Kakeya-type set in a
finite-dimensional space $K^d$ over a local field with finite residue field.
The construction is an adaptation of the ideas appearing in [12] and [13]. The
existence of measure-zero Kakeya-type sets over discrete valuation rings is
also discussed, giving an alternative construction to the one presented in [4]
over $\mathbb{F}_q[[t]]$.
