  We present a determination of the parton distribution functions of the proton
in which NLO and NNLO fixed-order calculations are supplemented by NLLx small-x
resummation. Deep inelastic structure functions are computed consistently at
NLO+NLLx or NNLO+NLLx, while for hadronic processes small-x resummation is
included only in the PDF evolution, with kinematic cuts introduced to ensure
the fitted data lie in a region where the fixed-order calculation of the hard
cross-sections is reliable. In all other respects, the fits use the same
methodology and are based on the same global dataset as the recent NNPDF3.1
analysis. We demonstrate that the inclusion of small-x resummation leads to a
quantitative improvement in the perturbative description of the HERA inclusive
and charm-production reduced cross-sections in the small x region. The impact
of the resummation in our fits is greater at NNLO than at NLO, because
fixed-order calculations have a perturbative instability at small x due to
large logarithms that can be cured by resummation. We explore the
phenomenological implications of PDF sets with small-x resummation for the
longitudinal structure function $F_L$ at HERA, for parton luminosities and LHC
benchmark cross-sections, for ultra-high energy neutrino-nucleus
cross-sections, and for future high-energy lepton-proton colliders such as the
LHeC.
