  In order to predict the black hole mass distributions at high redshift, we
need to understand whether very massive single stars ($M>40$ M$_\odot$) at low
metallicity $Z$ lose their hydrogen-rich envelopes, like their metal-rich
counterparts, or whether a binary companion is required to achieve this. To
test this, we undertake a deep spectroscopic search for binary companions of
the seven apparently single Wolf-Rayet (WR) stars in the Small Magellanic Cloud
(SMC; $Z \simeq 1/5 Z_\odot$). For each of them, we acquired six high-quality
VLT-UVES spectra spread over 1.5 years. By using the narrow N V lines in these
spectra, we monitor radial velocity (RV) variations to search for binary
motion. We find low RV variations between 6 and 23 km/s for the seven WR stars,
with a median standard deviation of $5$ km/s. Our Monte Carlo simulations imply
probabilities below ~5% for any of our target WR stars to have a binary
companion more massive than ~5 M$_\odot$ at orbital periods of less than a
year. We estimate that the probability that all our target WR stars have
companions with orbital periods shorter than 10 yr is below ~10$^{-5}$, and
argue that the observed modest RV variations may originate from intrinsic
atmosphere or wind variability. Our findings imply that metal-poor massive
stars born with $M \gtrsim 40$ M$_\odot$ can lose most of their hydrogen-rich
envelopes via stellar winds or eruptive mass loss, which strongly constrains
their initial mass - black hole mass relation. We also identify two of our
seven target stars (SMC AB1 and SMC AB11) as runaway stars with a peculiar
radial velocity of ~80 km/s. Moreover, with all five previously detected WR
binaries in the SMC exhibiting orbital periods of below 20 d, a puzzling
absence of intermediate-to-long-period WR binaries has emerged, with strong
implications for the outcome of massive binary interaction at low metallicity.
