Yes, if $X$ is a variety over an extension $K$ of $\mathbb Q_p$, then the $\ell$-adic cohomology spaces
$H^i(X,\mathbb Q_{\ell})$ are $\ell$-adic representations of $G_{K}$,
which give rise to Weil--Deligne representations.  (See Tate's Corvallis article,
for example.)  The resulting Weil--Deligne representation is conjectured to be
independent of the choice of $\ell$ (as long as $\ell \neq p$).  This is known
when $X$ has a smooth proper model over the ring of integers of $K$ (although
from the point of view of root numbers this case is not so interesting; in
this good reduction situation the associated Weil--Deligne representation
is unramified, so the local root number is 1, for the right choice of additive 
character).   It is also known when $X$ is an elliptic curve (in which case
$i = 1$ is the interesting choice; this gives the contragredient of the
Tate module).  

Actually computing root numbers is a non-trivial business, especially for
instances of very bad reduction at small primes.  For examples, see recent
work by Mazur and Rubin, and by the Dokchitsers.