(Abridged) Dynamical friction can be used to distinguish Newtonian gravity and modified Newtonian dynamics (MOND) because it works differently in these frameworks. This concept, however, has yet to be explored very much with MOND. Previous simulations showed weaker dynamical friction during major mergers for MOND than for Newtonian gravity with dark matter. Analytic arguments suggest the opposite for minor mergers. In this work, we verify the analytic predictions for MOND by high-resolution $N$-body simulations of globular clusters (GCs) moving in isolated ultra-diffuse galaxies (UDGs). We test the MOND analog of the Chandrasekhar formula for the dynamical friction proposed by Sanchez-Salcedo on a single GC. We also explore whether MOND allows GC systems of isolated UDGs to survive without sinking into nuclear star clusters. The simulations are run using the adaptive-mesh-refinement code Phantom of Ramses. The mass resolution is $20,M_odot$ and the spatial resolution $50,$pc. The GCs are modeled as point masses. Simulations including a single GC reveal that, as long as the apocenter of the GC is over about 0.5 effective radii, the Sanchez-Salcedo formula works excellently, with an effective Coulomb logarithm increasing with orbital circularity. Once the GC reaches the central kiloparsec, its sinking virtually stops, likely because of the core stalling mechanism. In simulations with multiple GCs, many of them sink toward the center, but the core stalling effect seems to prevent them from forming a nuclear star cluster. The GC system ends up with a lower velocity dispersion than the stars of the galaxy. By scaling the simulations, we extend these results to most UDG parameters, as long as these UDGs are not external-field dominated.
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