bounded language complete for NSPACE(log n)?
$\mathsf{L}=\mathsf{NL}$. Yes, there is an analog of Mahaney's Theorem for log-space. If there is a sparse set that is hard for $\mathsf{NL}$ under logspace reductions, then $\mathsf{L}=\mathsf{NL}$:

J.-Y. Cai and D. Sivakumar. Resolution of Hartmanis’ conjecture for NL-hard sparse sets. Theoret. Comp. Sci. 240(2):257–269, 2000.

van Melkebeek extends this to logspace bounded truth-table reductions, which is quite close to the state of the art for extensions of Mahaney's Theorem in the polytime world as well:

D. van Melkebeek. Deterministic and Randomized Bounded Truth-Table Reductions of P, NL, and L to Sparse Sets. J. Comput. Syst. Sci. 57(2):213-232, 1998.

(I think the reversed dates are simply because these are the journal publications rather than the corresponding conference publications.)