Indivisible ultrametric spaces

A metric space is indivisible if for any partition of it into finitely many pieces one piece contains an isometric copy of the whole space. Continuing our investigation of indivisible metric spaces [C. Delhommé, C. Laflamme, M. Pouzet, N. Sauer, Divisibility of countable metric spaces, European J. Combin. 28 (2007) 1746–1769], we show that a countable ultrametric space is isometrically embeddable into an indivisible ultrametric space if and only if it does not contain a strictly increasing sequence of balls.