  For a finite extension $F$ of ${\mathbf Q}_p$, Drinfeld defined a tower of
coverings of ${\mathbb P}^1\setminus {\mathbb P}^1(F)$ (the Drinfeld
half-plane). For $F = {\mathbf Q}_p$, we describe a decomposition of the
$p$-adic geometric \'etale cohomology of this tower analogous to Emerton's
decomposition of completed cohomology of the tower of modular curves. A crucial
ingredient is a finitness theorem for the arithmetic \'etale cohomology modulo
$p$ which is shown by first proving, via a computation of nearby cycles, that
this cohomology has finite presentation. This last result holds for all $F$;
for $F\neq {\mathbf Q}_p$, it implies that the representations of ${\rm
GL}_2(F)$ obtained from the cohomology of the Drinfeld tower are not admissible
contrary to the case $F = {\mathbf Q}_p$.
