Step 1: Recall Jahn–Teller theorem.
Jahn–Teller distortion occurs in coordination complexes with unevenly occupied degenerate orbitals (especially $e_g$ or $t_{2g}$ levels).
Step 2: Check electronic configurations.
(A) Co(III): d$^6$ low-spin → $t_{2g}^6e_g^0$ (no degeneracy) → no distortion.
(B) Ni(II): d$^8$ → in octahedral field → $t_{2g}^6e_g^2$, unevenly filled $e_g$ orbitals → Jahn–Teller active.
(C) Mn(II): d$^5$ high-spin → symmetric half-filled → no distortion.
(D) Co(III): d$^6$ low-spin → $t_{2g}^6$ → no distortion.
Step 3: Conclusion.
The only complex showing Jahn–Teller distortion is [NiF$_6$]$^{2-$ (Option B)}.