  We present a first determination of the nuclear parton distribution functions
(nPDF) based on the NNPDF methodology: nNNPDF1.0. This analysis is based on
neutral-current deep-inelastic structure function data and is performed up to
NNLO in QCD calculations with heavy quark mass effects. For the first time in
the NNPDF fits, the $\chi^2$ minimization is achieved using stochastic gradient
descent with reverse-mode automatic differentiation (backpropagation). We
validate the robustness of the fitting methodology through closure tests,
assess the perturbative stability of the resulting nPDFs, and compare them with
other recent analyses. The nNNPDF1.0 distributions satisfy the boundary
condition whereby the NNPDF3.1 proton PDF central values and uncertainties are
reproduced at $A=1$, which introduces important constraints particularly for
low-$A$ nuclei. We also investigate the information that would be provided by
an Electron-Ion Collider (EIC), finding that EIC measurements would
significantly constrain the nPDFs down to $x\simeq 5\times 10^{-4}$. Our
results represent the first-ever nPDF determination obtained using a Monte
Carlo methodology consistent with that of state-of-the-art proton PDF fits, and
provide the foundation for a subsequent global nPDF analyses including also
proton-nucleus data.
