Are generically stable types stationary? Is every type that is generically stable over a model $M$, stationary over $M$? (Without any assumption on $T$.)
Some definitions:
A global type $p(x)$ is generically stable over $M$ if 1 and 2 below hold


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*$p(x)$ is definable over $M$, that is, for every formula $\varphi(x,y)\in L$ there is a formula $\vartheta(y)\in L(M)$ such that $\varphi(x,a)\in p$ iff $\vartheta(a)$. 

*$p(x)$ is finitely satisfiable in $M$, that is, for every formula $\varphi(x)\in p$ there is an $a\in M$ such that $\varphi(a)$.
I say that $p(x)$ is stationary over $M$ (I don't know if this is standard terminology) if $p(x)$ is the unique global type invariant over $M$ that extends $p_{\restriction M}(x)$.

In Proposition 1 (iv) on p. 4 of this paper, Pillay and Tanovic prove that in the context of an arbitrary theory $T$, if $p$ is a generically stable type over $A$, then $p$ is the unique global non-forking extension of $p|_A$. It follows that $p$ is the unique global $A$-invariant extension of $p|_A$, since $A$-invariant types do not fork over $A$. 
The only problem is that the definition of "generically stable type" in that paper differs from yours. Pillay and Tanovic say that a global type $p$ is generically stable over $A$ if $p$ is $A$-invariant and for any Morley sequence $I$ in $p$ over $A$ and any formula $\phi(x)$ with parameters from the monster model, $\phi(I)$ is a finite or cofinite subset of $I$. 
The Pillay-Tanovic definition implies yours (this is Proposition 1 (ii)) and is equivalent to yours if $T$ is NIP (see Theorem 2.29 in Simon's A Guide to NIP Theories), and the proof of this implication uses NIP in a serious way (but I don't have a counterexample in a theory with IP in mind, and I would like to see one!). So it seems like the Pillay-Tanovic definition is probably the right one to use outside the NIP setting. 
