  We continue to investigate the $k$ nearest neighbour ($k$-NN) learning rule
in complete separable metric spaces. Thanks to the results of C\'erou and
Guyader (2006) and Preiss (1983), this rule is known to be universally
consistent in every such metric space that is sigma-finite dimensional in the
sense of Nagata. Here we show that the rule is strongly universally consistent
in such spaces in the absence of ties. Under the tie-breaking strategy applied
by Devroye, Gy\"{o}rfi, Krzy\.{z}ak, and Lugosi (1994) in the Euclidean
setting, we manage to show the strong universal consistency in non-Archimedian
metric spaces (that is, those of Nagata dimension zero). Combining the theorem
of C\'erou and Guyader with results of Assouad and Quentin de Gromard (2006),
one deduces that the $k$-NN rule is universally consistent in metric spaces
having finite dimension in the sense of de Groot. In particular, the $k$-NN
rule is universally consistent in the Heisenberg group which is not
sigma-finite dimensional in the sense of Nagata as follows from an example
independently constructed by Kor\'anyi and Reimann (1995) and Sawyer and
Wheeden (1992).
