We report transverse-field (TF) muon spin rotation experiments on single crystals of the topological superconductor Sr$_x$Bi$_2$Se$_3$ with nominal concentrations $x=0.15$ and $0.18$ ($T_c sim 3$ K). The TF spectra ($B= 10$ mT), measured after cooling to below $T_c$ in field, did not show any additional damping of the muon precession signal due to the flux line lattice within the experimental uncertainty. This puts a lower bound on the magnetic penetration depth $lambda geq 2.3 ~mu$m. However, when we induce disorder in the vortex lattice by changing the magnetic field below $T_c$ a sizeable damping rate is obtained for $T rightarrow 0$. The data provide microscopic evidence for a superconducting volume fraction of $sim 70~ %$ in the $x=0.18$ crystal and thus bulk superconductivity.
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