We report results of muon spin relaxation and rotation ($mu$SR) experiments on the spin-liquid candidate~YbMgGaO$_{4}$. No static magnetism $gtrsim 0.003mu_B$ per Yb ion, ordered or disordered, is observed down to 22~mK, a factor of two lower in temperature than previous measurements. Persistent (temperature-independent) spin dynamics are observed up to 0.20~K and at least 1~kOe, thus extending previous zero-field $mu$SR results over a substantial region of the $H$-$T$ phase diagram. Knight shift measurements in a 10-kOe transverse field reveal two lines with nearly equal amplitudes. Inhomogeneous muon depolarization in a longitudinal field, previously characterized by stretched-exponential relaxation due to spatial inhomogeneity, is fit equally well with two exponentials, also of equal amplitudes. We attribute these results to two interstitial muon sites in the unit cell, rather than disorder or other spatial distribution. Further evidence for this attribution is found from agreement between the ratio of the two measured relaxation rates and calculated mean-square local Yb$^{3+}$ dipolar fields at candidate muon sites. Zero-field data can be understood as a combination of two-exponential dynamic relaxation and quasistatic nuclear dipolar fields.
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