  We report muon spin relaxation ($\mu$SR) and $^{69,71}$Ga nuclear quadrupolar
resonance (NQR) local-probe investigations of the kagom\'e compound
Pr$_3$Ga$_5$SiO$_{14}$. Small quasi-static random internal fields develop below
40 K and persist down to our base temperature of 21 mK. They originate from
hyperfine-enhanced $^{141}$Pr nuclear magnetism which requires a non-magnetic
Pr$^{3+}$ crystal-field (CF) ground state. Besides, we observe a broad maximum
of the relaxation rate at $\simeq 10$ K which we attribute to the population of
the first excited magnetic CF level. Our results yield a Van-Vleck paramagnet
picture, at variance with the formerly proposed spin-liquid ground state.
