  We study the notion of $p$-quasihomotopy in Newtonian classes of mappings and
link it to questions concerning lifts of Newtonian maps, under the assumption
that the target space is nonpositively curved. Using this connection we prove
that every $p$-quasihomotopy class of Newtonian maps contains a minimizer of
the $p$-energy if the target has hyperbolic fundamental group.
