  The origin of Galactic cosmic rays remains a matter of debate, but supernova
remnants are commonly considered to be the main place where high-energy cosmic
rays are accelerated. Nevertheless, current models predict cosmic-ray spectra
that do not match observations and the efficiency of the acceleration mechanism
is still undetermined. On the other hand, the contribution of other kinds of
sources to the Galactic cosmic-ray population is still unclear, and merits
investigation. In this work we explore a novel mechanism through which
microquasars might produce cosmic rays. In this scenario, microquasar jets
generate relativistic neutrons, which escape and decay outside the system;
protons and electrons, created when these neutrons decay, escape to the
interstellar medium as cosmic rays. The most promising scenarios arise in
extremely luminous systems ($L_\mathrm{jet} \sim 10^{40}\,\mathrm{erg \,
s}^{-1}$), in which the fraction of jet power deposited in cosmic rays can
reach $\sim 0.001$. Slow jets ($\Gamma \lesssim 2$, where $\Gamma$ is the bulk
Lorentz factor) favour neutron production. The resulting cosmic-ray spectrum is
similar for protons and electrons, which share the power in the ratio given by
neutron decay. The spectrum peaks at roughly half the minimum energy of the
relativistic protons in the jet; it is soft (spectral index $\sim 3$) above
this energy, and almost flat below. Values of spectral index steeper than $2$
are possible for cosmic rays in our model and these indeed agree with those
required to explain the spectral signatures of Galactic cosmic rays, although
only the most extreme microquasars provide power comparable to that of a
typical supernova remnant. The mechanism explored in this work may provide
stronger and softer cosmic-ray sources in the early Universe, and therefore
contribute to the heating and reionisation of the intergalactic medium.
