  In the quasi-1D heavy-fermion system YbNi$_4$(P$_{1-x}$As$_x$)$_2$ the
presence of a ferromagnetic (FM) quantum critical point (QCP) at $x_c$ $\approx
0.1$ with unconventional quantum critical exponents in the thermodynamic
properties has been recently reported. Here, we present muon-spin relaxation
($\mu$SR) experiments on polycrystals of this series to study the magnetic
order and the low energy 4$f$-electronic spin dynamics across the FM QCP. The
zero field $\mu$SR measurements on pure YbNi$_4$(P$_{2}$ proved static long
range magnetic order and suggested a strongly reduced ordered Yb moment of
about 0.04$\mu_B$. With increasing As substitution the ordered moment is
reduced by half at $x = 0.04$ and to less than 0.005 $\mu_B$ at $x=0.08$. The
dynamic behavior in the $\mu$SR response show that magnetism remains
homogeneous upon As substitution, without evidence for disorder effect. In the
paramagnetic state across the FM QCP the dynamic muon-spin relaxation rate
follows 1/$T_{1}T\propto T^{-n}$ with $1.01 \pm 0.04 \leq n \leq 1.13 \pm
0.06$. The critical fluctuations are very slow and are even becoming slower
when approaching the QCP.
