  The in-plane London penetration depth, $\Delta\lambda(T)$, was measured using
a tunnel diode resonator in single crystals of Ba$_{1-x}$K$_x$Fe$_2$As$_2$ for
five doping levels $x$ ranging from underdoped, $T_c$=11 K ($x$=0.17), to
optimally doped, $T_c=$38 K ($x_\textmd{\scriptsize opt}$= 0.35). In the
optimally doped samples, $\Delta\lambda(T)$ shows exponential saturation in $T
\to 0$ limit suggesting a fully gapped superconductivity. The lowest-$T_c$
samples show much stronger non-exponential variation of $\Delta\lambda(T)$.
Fitting the data to a power-law, $\Delta\lambda(T)= AT^n$, reveals a monotonic
decrease of the exponent $n$ with $x$ towards the underdoped edge of the
superconducting dome. Comparison with $n \approx 1.2$ reported in KFe$_2$As$_2$
($T_c$=3.5 K, $x=$ 1), suggests a dome-like variation of $n$ with $x$, implying
an evolution of the topology of the superconducting gap from full and isotropic
in the center of the dome towards strongly anisotropic and, eventually, nodal
at the dome edges.
