  Identifying the uniqueness of FeP-based superconductors may shed new lights
on the mechanism of superconductivity in iron-pnictides. Here, we report
nuclear magnetic resonance(NMR) studies on LiFeP and LiFeAs which have the same
crystal structure but different pnictogen atoms. The NMR spectrum is sensitive
to inhomogeneous magnetic fields in the vortex state and can provide the
information on the superconducting pairing symmetry through the temperature
dependence of London penetration depth $\lambda_L$. We find that $\lambda_L$
saturates below $T \sim 0.2$ $T_c$ in LiFeAs, where $T_c$ is the
superconducting transition temperature, indicating nodeless superconducting
gaps. Furthermore, by using a two-gaps model, we simulate the temperature
dependence of $\lambda_L$ and obtain the superconducting gaps of LiFeAs, as
$\Delta_1 = 1.2$ $k_B T_c$ and $\Delta_2 = 2.8$ $k_B T_c$, in agreement with
previous result from spin-lattice relaxation. For LiFeP, in contrast, the
London penetration depth $\lambda_L$ does not show any saturation down to $T
\sim 0.03 $ $T_c$, indicating nodes in the superconducting energy gap function.
Finally, we demonstrate that the strong spin fluctuations with diffusive
characteristics exist in LiFeP, as in some cuprate high temperature
superconductors.
