  We compute the leading order (LO) $qg\to q \gamma$ and next-to-leading order
(NLO) $gg\to q{\bar q} \gamma$ contributions to inclusive photon production in
proton-proton (p+p) collisions at the LHC. These channels provide the dominant
contribution at LO and NLO for photon transverse momenta $k_{\gamma\perp}$
corresponding to momentum fractions of $x\leq 0.01$ in the colliding protons.
Our computations, performed in the dilute-dense framework of the Color Glass
Condensate effective field theory (CGC EFT), show that the NLO contribution
dominates at small-$x$ because it is sensitive to $k_\perp$-dependent
unintegrated gluon distributions in both of the protons. We predict a maximal
$10\%$ modification of the cross section at low $k_{\gamma\perp}$ as a direct
consequence of the violation of $k_\perp$-factorization. The coherence effects
responsible for this modification are enhanced in nuclei and can be identified
from inclusive photon measurements in proton-nucleus collisions. We provide
numerical results for the isolated inclusive photon cross section for
$k_{\gamma\perp}\leq 20$ GeV in p+p collisions that can be tested in the future
at the LHC.
