  The mechanism of unconventional superconductivity in iron-based
superconductors (IBSs) is one of the most intriguing questions in current
materials research. Among non-oxide IBSs, (Ba$_{1-x}$K$_x$)Fe$_2$As$_2$ has
been intensively studied because of its high superconducting transition
temperature and fascinating evolution of the superconducting gap structure from
being fully isotropic at optimal doping (\(x\approx\)0.4) to becoming nodal at
$x > $0.8. Although this marked evolution was identified in several independent
experiments, there are no details of the gap evolution to date because of the
lack of high-quality single crystals covering the entire K-doping range of the
superconducting dome. We conducted a systematic study of the London penetration
depth, $\lambda (T)$, across the full phase diagram for different
concentrations of point-like defects introduced by 2.5 MeV electron
irradiation. Fitting the low-temperature variation with the power law, $\Delta
\lambda \sim T^{n}$, we find that the exponent $n$ is the highest and $T_c$
suppression rate with disorder is the smallest at optimal doping, and they
evolve with doping being away from optimal, which is consistent with increasing
gap anisotropy, including an abrupt change around $x\simeq 0.8$, indicating the
onset of nodal behavior. Our analysis using a self-consistent $t$-matrix
approach suggests the ubiquitous and robust nature of s$_{\pm}$ pairing in IBSs
and argues against a previously suggested transition to a $d-$wave state near
$x=1$ in this system.
