  We present results for the isovector electromagnetic form factors of the
nucleon computed on the CLS ensembles with $N_f=2+1$ flavors of
$\mathcal{O}(a)$-improved Wilson fermions and an $\mathcal{O}(a)$-improved
vector current. The analysis includes ensembles with four lattice spacings and
pion masses ranging from 130 MeV up to 350 MeV and mainly targets the low-$Q^2$
region. In order to remove any bias from unsuppressed excited-state
contributions, we investigate several source-sink separations between 1.0 fm
and 1.5 fm and apply the summation method as well as explicit two-state fits.
The chiral interpolation is performed by applying covariant chiral perturbation
theory including vector mesons directly to our form factor data, thus avoiding
an auxiliary parametrization of the $Q^2$ dependence. At the physical point, we
obtain $\mu=4.71(11)_{\mathrm{stat}}(13)_{\mathrm{sys}}$ for the nucleon
isovector magnetic moment, in good agreement with the experimental value and
$\langle
r_\mathrm{M}^2\rangle~=~0.661(30)_{\mathrm{stat}}(11)_{\mathrm{sys}}\,~\mathrm{fm}^2$
for the corresponding square-radius, again in good agreement with the value
inferred from the $ep$-scattering determination [Bernauer et~al., Phys. Rev.
Lett., 105, 242001 (2010)] of the proton radius. Our estimate for the isovector
electric charge radius, $\langle r_\mathrm{E}^2\rangle =
0.800(25)_{\mathrm{stat}}(22)_{\mathrm{sys}}\,~\mathrm{fm}^2$, however, is in
slight tension with the larger value inferred from the aforementioned
$ep$-scattering data, while being in agreement with the value derived from the
2018 CODATA average for the proton charge radius.
