  Arveson's extension theorem guarantees that every completely positive map
defined on an operator system can be extended to a completely positive map
defined on the whole C*-algebra containing it. An analogous statement where
complete positivity is replaced by positivity is known to be false. A natural
question is whether extendibility could still hold for positive maps satisfying
stronger conditions, such as being unital and norm 1. Here we provide three
counterexamples showing that positive norm-one unital maps defined on an
operator subsystem of a matrix algebra cannot be extended to a positive map on
the full matrix algebra. The first counterexample is an unextendible positive
unital map with unit norm, the second counterexample is an unextendible
positive unital isometry on a real operator space, and the third counterexample
is an unextendible positive unital isometry on a complex operator space.
