  In contrast to an infinite family of explicit examples of two-dimensional
$p$-harmonic functions obtained by G.Aronsson in the late 80s, there is very
little known about the higher-dimensional case. In this paper, we show how to
use isoparametric polynomials to produce diverse examples of $p$-harmonic and
biharmonic functions. Remarkably, for some distinguished values of $p$ and the
ambient dimension $n$ this yields first examples of rational and algebraic
$p$-harmonic functions. Moreover, we show that there are no $p$-harmonic
polynomials of the isoparametric type. This supports a negative answer to a
question proposed in 1980 by J. Lewis.
