In the quasi-one-dimensional heavy-fermion system YbNi₄(P_1−xAs_x)₂ the presence of a ferromagnetic (FM) quantum critical point (QCP) at x_c ≈ 0.1 with unconventional quantum critical exponents in the thermodynamic properties has been recently reported. Here, we present muon-spin relaxation (μSR) experiments on polycrystals of this series to study the magnetic order and the low-energy 4f-electronic spin dynamics across the FM QCP. The zero-field μSR measurements on pure YbNi₄P₂ proved static long-range magnetic order and suggested a strongly reduced ordered Yb moment of about 0.04μ_B. With increasing As substitution, the ordered moment is reduced by half at x = 0.04 and to less than 0.005μ_B at x = 0.08. The dynamic behavior in the μSR response shows that magnetism remains homogeneous upon As substitution, without evidence of a disorder effect. In the paramagnetic state across the FM QCP the dynamic muon-spin relaxation rate follows 1/T1T ∝ T⁻ⁿ with 1.01 ± 0.04 ≤ n ≤ 1.13 ± 0.06. The critical fluctuations are very slow and become even slower when approaching the QCP.