  We show that for k a perfect field of characteristic p, there exist
endomorphisms of the completed algebraic closure of k((t)) which are not
bijective. As a corollary, we resolve a question of Fargues and Fontaine by
showing that for p a prime and C_p a completed algebraic closure of Q_p, there
exist closed points of the Fargues-Fontaine curve associated to C_p whose
residue fields are not (even abstractly) isomorphic to C_p as topological
fields.
