  We give a new definition of a $p$-adic $L$-function for a mixed signature
character of a real quadratic field and for a nontrivial ray class character of
an imaginary quadratic field. We then state a $p$-adic Stark conjecture for
this $p$-adic $L$-function. We prove our conjecture in the case when $p$ is
split in the imaginary quadratic field by relating our construction to Katz's
$p$-adic $L$-function. We also provide numerical evidence for our conjecture in
three examples.
