Many extensions of the Standard Model of particle physics explain the dominance of matter over antimatter in our Universe by neutrinos being their own antiparticles. This would imply the existence of neutrinoless double-beta decay, which is an extremely rare lepton-number-violating radioactive decay process whose detection requires the utmost background suppression. Among the programmes that aim to detect this decay, the GERDA Collaboration is searching for neutrinoless double-beta decay of Ge-76 by operating bare detectors, made of germanium with an enriched Ge-76 fraction, in liquid argon. After having completed Phase I of data taking, we have recently launched Phase II. Here we report that in GERDA Phase II we have achieved a background level of approximately 10(-3) counts keV(-1) kg(-1) yr(-1). This implies that the experiment is background-free, even when increasing the exposure up to design level. This is achieved by use of an active veto system, superior germanium detector energy resolution and improved background recognition of our new detectors. No signal of neutrinoless double-beta decay was found when Phase I and Phase II data were combined, and we deduce a lower-limit half-life of 5.3 x 10(25) years at the 90 per cent confidence level. Our half-life sensitivity of 4.0 x 10(25) years is competitive with the best experiments that use a substantially larger isotope mass. The potential of an essentially background-free search for neutrinoless double-beta decay will facilitate a larger germanium experiment with sensitivity levels that will bring us closer to clarifying whether neutrinos are their own antiparticles.