It's not 0.8^10 anyway, because there aren't 10 neighbours.

There are neighbours in both directions in each dimension, i.e. 20 neighbours in 10 dimensions if you're allowed to wrap around the edges of the lattice.

With wrapping, I think the probability is 0.2 × (0.8^20) ≈ 0.002306 ≈ 0.23%.

If you're not allowed to wrap at the edges, a unformly random point has 2/N probability of having one neighbour on each dimension independently, and (N-2)/N probability of having two. With D dimensions, that's on average D(2N-2)/N neighbours.

With D = 10, N = 5, I think it's 16 neighbours on average, with a distribution from 10 to 20.

That makes the probability of landing on a mountain peak lower than ~2% and higher than ~0.23%. (Not 0.2 × (0.8^16) though, due to the distribution.)