nt.number theory - Mod l local Galois representations (l different from p)

My question is referred to the statement and proof of Prop. 2.4 of Diamond's
article "An extension of Wiles' Results", in Modular Forms and Fermat Last
Theorem, page 479.

More precisely: fix $l$ and $p$ two distinct primes, with $l$ odd. Let $sigma$ be an
irreducible, continuous, degree 2 representation of the absolute Galois group
$G_{p}$ of $Q_{p}$, with coefficients in $k$, an algebraic closure of the finite
field with $l$ elements.
Proposition 2.4 states that if the restriction of $sigma$ to the inertia
subgroup of $G_{p}$ is irreducible and $p$ is odd, then $sigma$ is isomorphic to the
representation induced from a character of the Galois group of a quadratic
ramified extension $M$ of $Q_{p}$.
The proof given works if the restriction of $sigma$ to the wild inertia of $G_{p}$ is
reducible (I think there's a typo in the first line of the proof). What if
$sigma$ is irreducible on wild inertia (and $p$ is always odd)? It seems to me that this case is not covered in the proof of the Proposition, but maybe I'm not seeing something obvious.. If such a representation exists, it cannot be induced from a quadratic extension $M$ as above, so how does it fit in the description given by the Proposition? Can one say something about such a representation (for example something about its projective image?).

Thanks

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