  We prove estimates of a $p$-harmonic measure, $p \in (n-m, \infty]$, for sets
in $\mathbf{R}^n$ which are close to an $m$-dimensional hyperplane $\Lambda
\subset \mathbf{R}^n$, $m \in [0,n-1]$. Using these estimates, we derive
results of Phragm\'en-Lindel\"of type in unbounded domains $\Omega \subset
\mathbf{R}^n\setminus \Lambda$ for $p$-subharmonic functions. Moreover, we give
local and global growth estimates for $p$-harmonic functions, vanishing on sets
in $\mathbf{R}^n$, which are close to an $m$-dimensional hyperplane.
