  Quantum state manipulation with gates based on geometric phases acquired
during cyclic operations promises inherent fault-tolerance and resilience to
local fluctuations in the control parameters. Here we create a general
non-Abelian and non-adiabatic holonomic gate acting in the $(\ket{0},\ket{2})$
subspace of a three-level transmon fabricated in a fully coplanar design.
Experimentally, this is realized by simultaneously coupling the first two
transitions by microwave pulses with amplitudes and phases defined such that
the condition of parallel transport is fulfilled. We demonstrate the creation
of arbitrary superpositions in this subspace by changing the amplitudes of the
pulses and the relative phase between them. We use two-photon pulses acting in
the holonomic subspace to reveal the coherence of the state created by the
geometric gate pulses and to prepare different superposition states. We also
test the action of holonomic NOT and Hadamard gates on superpositions in the
$(\ket{0},\ket{2})$ subspace.
