ABSTRACT We present a study designed to measure the average Lyman-continuum escape fraction (〈fesc〉) of star-forming galaxies at z ≃ 3.5. We assemble a sample of 148 galaxies from the VANDELS spectroscopic survey at 3.35 ≤ zspec ≤ 3.95, selected to minimize line-of-sight contamination of their photometry. For this sample, we use ultra-deep, ground-based, U-band imaging and Hubble Space Telescope V-band imaging to robustly measure the distribution of $\mathcal {R_{\rm obs}}\, =(L_{\rm LyC}/L_{\rm UV})_{\rm obs}$. We then model the $\mathcal {R_{\rm obs}}$ distribution as a function of 〈fesc〉, carefully accounting for attenuation by dust, the intergalactic medium and the circumgalactic medium. A maximum likelihood fit to the $\mathcal {R_{\rm obs}}$ distribution returns a best-fitting value of $〈 f_{\rm esc}〉 =0.07^{+0.02}_{-0.02}$, a result confirmed using an alternative Bayesian inference technique (both techniques exclude 〈fesc〉 = 0.0 at >3σ). By splitting our sample in two, we find evidence that 〈fesc〉 is positively correlated with Ly α equivalent width (Wλ(Ly α)), with high and low Wλ(Lyα) subsamples returning values of $〈 f_{\rm esc}〉 =0.12^{+0.06}_{-0.04}$ and $〈 f_{\rm esc} 〉 =0.02^{+0.02}_{-0.01}$, respectively. In contrast, we find evidence that 〈fesc〉 is anticorrelated with intrinsic UV luminosity and UV dust attenuation; with low UV luminosity and dust attenuation subsamples both returning best fits in the range 0.10 ≤ 〈fesc〉 ≤ 0.22. We do not find a clear correlation between fesc and galaxy stellar mass, suggesting stellar mass is not a primary indicator of fesc. Although larger samples are needed to further explore these trends, our results suggest that it is entirely plausible that the low dust, low-metallicity galaxies found at z ≥ 6 will display the 〈fesc〉 ≥ 0.1 required to drive reionization.