  The canonical generalizations of two classical norms on Besov spaces are
shown to be equivalent even in the case of non-linear Besov spaces, that is,
function spaces consisting of functions taking values in a metric space and
equipped with some Besov-type topology. The proofs are based on atomic
decomposition techniques and metric embeddings. Additionally, we provide
embedding results showing how non-linear Besov spaces embed into non-linear
$p$-variation spaces and vice versa. We emphasize that we neither assume the
UMD property of the involved spaces nor their separability.
