  We present updated results for the form factors of the kaon semileptonic
$(K_{\ell 3})$ decay process calculated with $N_f = 2 + 1$ nonperturbatively
$O(a)$-improved Wilson quark action and Iwasaki gauge action at the physical
point on large volumes of more than (10 fm)$^4$. In addition to our previous
calculation at the lattice spacing $a = 0.085$ fm, we perform a calculation at
the second lattice spacing of $0.063$ fm. Using the results for the form
factors extracted from 3-point functions with the local and also conserved
vector currents at the two lattice spacings, continuum extrapolation and
interpolation of the momentum transfer are carried out simultaneously to obtain
the value of the form factor $f_+(0)$ at the zero momentum transfer in the
continuum limit. After investigation of stability of $f_+(0)$ against several
fit forms and different data, we obtain $f_+(0) = 0.9615(10)(^{+47}_{\
-3})(5)$, where the first, second, and third errors are statistical, systematic
errors from choice of the fit forms and isospin breaking effect, respectively.
Combining our value of $f_+(0)$ and experimental input of the $K_{\ell 3}$
decay, one of the Cabibbo-Kobayashi-Maskawa matrix elements $|V_{us}|$ is
determined as $|V_{us}| = 0.2252(^{\ +5}_{-12})$, whose error contains the
experimental one as well as that in the lattice calculation. This value is
reasonably consistent with the ones determined from recent lattice QCD results
of $f_+(0)$ and also the one determined through the kaon leptonic decay
process. We observe some tension between our value and $|V_{us}|$ evaluated
from the unitarity of the CKM matrix with $|V_{ud}|$, while it depends on the
size of the error of $|V_{ud}|$. It is also found that $|V_{us}|$ determined
with our phase space integrals through six $K_{\ell 3}$ decay processes is
consistent with the above one using $f_+(0)$.
