  This paper presents studies of Bose-Einstein correlations (BEC) in
proton-proton collisions at a centre-of-mass energy of 13 TeV, using data from
the ATLAS detector at the CERN Large Hadron Collider. Data were collected in a
special low-luminosity configuration with a minimum-bias trigger and a
high-multiplicity track trigger, accumulating integrated luminosities of 151
$\mu$b$^{-1}$ and 8.4 nb$^{-1}$ respectively. The BEC are measured for pairs of
like-sign charged particles, each with $|\eta|$ < 2.5, for two kinematic
ranges: the first with particle $p_T$ > 100 MeV and the second with particle
$p_T$ > 500 MeV. The BEC parameters, characterizing the source radius and
particle correlation strength, are investigated as functions of
charged-particle multiplicity (up to 300) and average transverse momentum of
the pair (up to 1.5 GeV). The double-differential dependence on
charged-particle multiplicity and average transverse momentum of the pair is
also studied. The BEC radius is found to be independent of the charged-particle
multiplicity for high charged-particle multiplicity (above 100), confirming a
previous observation at lower energy. This saturation occurs independent of the
transverse momentum of the pair.
