  We present an investigation of the electromagnetic pion form factor,
$F_\pi(Q^2)$, at small values of the four-momentum transfer $Q^2$ ($\lesssim
0.25$ GeV$^2$), based on the gauge configurations generated by European Twisted
Mass Collaboration with $N_f = 2$ twisted-mass quarks at maximal twist
including a clover term. Momentum is injected using non-periodic boundary
conditions and the calculations are carried out at a fixed lattice spacing ($a
\simeq 0.09$ fm) and with pion masses equal to its physical value, 240 MeV and
340 MeV. Our data are successfully analyzed using Chiral Perturbation Theory at
next-to-leading order in the light-quark mass. For each pion mass two different
lattice volumes are used to take care of finite size effects. Our final result
for the squared charge radius is $\langle r^2 \rangle_\pi = 0.443~(29)$ fm$^2$,
where the error includes several sources of systematic errors except the
uncertainty related to discretization effects. The corresponding value of the
SU(2) chiral low-energy constant $\overline{\ell}_6$ is equal to
$\overline{\ell}_6 = 16.2 ~ (1.0)$.
