Ok, I think a few things need to be teased apart to answer this question. As I see it, an answer to this question must first address the following questions:
- What is Philosophy?
- What is Logic?
I'll address them in turn (though my answer to the first won't be very satisfying to most, I suspect).
What is Philosophy?
This is really my least favorite "philosophical" question. I think it's a question that should, largely, be written off as a boondoggle. There are no hard and fast borders, at least as it is practiced now. First, there is the Analytic/Continental divide within philosophy (the distinction is imperfect but it's the only one I know of). I'm mostly familiar with the analytic tradition, so my answer will be restricted to that tradition.
In contemporary analytic philosophy, many fields are virtually indistinguishable from non-philosophical counterparts. There are philosophical logicians who do work indistinguishable from that of mathematical logicians. There are philosophers of mind who work in tandem with cognitive scientists and so, quite literally, do the same thing. Does this mean cognitive science is philosophy? No, but it doesn't mean that philosophy of mind is cognitive science either.
Against the perspective given in Chris Sunami's answer, I'll note that there are many who follow W.V.O. Quine in holding that "philosophy is continuous with science". This is sometimes called "naturalism". Here is Quine on the matter (from this IEP article):
What distinguishes between the ontological philosopher’s concerns and
…[zoology, botany, and physics] is only breadth of categories. Given
physical objects in general, the natural scientist is the man to
decide about wombats and unicorns. Given classes…it is the
mathematician to say whether in particular there are any even prime
numbers…On the other hand it is the scrutiny of this uncritical
acceptance of the realm of physical objects itself, or of classes,
etc., that devolves upon ontology. (Quine 1960, 275)
From the same article:
The basic conception of philosophy and philosophical practice that
informs [Quine's] discussion of science is commonly know as naturalism, a
view that recommends the “abandonment of the goal of a first
philosophy prior to natural science” (1981, 67), which further
involves a “readiness to see philosophy as natural science trained
upon itself and permitted free use of scientific findings” (1981, 85)
and lastly, recognizes that “…it is within science itself, and not in
some prior philosophy, that reality is to be identified and described”
I think that the best you could say is that what distinguishes philosophy from other disciplines is methodology -- but even that is tenuous and hard to specify. I like to think of it somewhat like how Justice Stewart characterized pornography, "I can't tell you what it is, but I know it when I see it" (paraphrased).
What is Logic?
First, as I noted in my comment on Chris Sunami's answer, there is a distinction between Philosophical Logic and Mathematical Logic. (See also Peter Smith's answer I linked to in my comment above.) Again, the boundaries here are fuzzy but I think the distinction gets at something.
(As an aside, what you refer to as "informal logic" is typically taught in critical thinking classes. It is rarely studied at a research level, however, and is typically only taught at an undergraduate level. This is not to say that there aren't some philosophers who do research in this area, just that it isn't what most would think of when they think of "logic".)
Now, what differentiates Philosophical Logic from Mathematical Logic? Well, Philosophical Logic tends to be focused on certain applications of formal logic to traditional philosophical questions. For instance, W.V.O. Quine in his landmark article on ontology, "On What There Is", used the formalism of first-order logic to make tractable questions of existence. In this article, Quine gave us his famous dictum "to be is to be the value of a variable" -- effectively reducing questions of existence to questions of the range of our first-order quantifiers. (Others have expanded this criterion to include the values of higher-order variables, notably interpreting second-order variables as having property-like entities as values.)
Another example is the study of modal logic. Within philosophical traditions modal logic is commonly studied as the logic of metaphysical necessity/possibility (S5 modal logic, usually). When doing modal logic with its traditional Kripke Semantics, there is an accessibility relation between frames. In S5 modal logic this gets ignored since S5 accessibility is an equivalence relation and so, on the assumption that only one possible world is actual, every world accesses every other world. Sometimes this is referred to as absolute modality. Weaker modal logics are typically understood as capturing some notion of relative possibility. Other applications (discussed in the SEP article linked above) are as follows:
□ It is necessary that ..
◊ It is possible that …
O It is obligatory that …
P It is permitted that …
F It is forbidden that …
G It will always be the case that …
F It will be the case that …
H It has always been the case that …
P It was the case that …
Bx x believes that …
By contrast, modal logic within mathematical logic has entirely different interpretations. For example, there are algebraic approaches, where, e.g., S4 modal logic is interpreted as an interior algebra.
Both philosophers and mathematicians are interested in the application of modal logic to the study of provability, under the heading of so-called "provability logics". Additionally the work of Joel David Hamkins (a mainstay at Mathoverflow) is of considerable interest to both mathematicians and philosophers. His discussion of the Set-theoretic Multiverse is influential among those who favor a pluralistic conception of mathematical truth.
Additionally, it's common to group set theory and model theory under the heading of "mathematical logic". Both branches are of massive interest to philosophers and mathematicians (and linguists and computer scientists, etc.). If I had to characterize the distinction between these two fields I would say that the philosophers tend to be more interested in drawing "big picture" conclusions about the nature of, e.g., mathematical truth whereas mathematicians tend to be more concerned with what can be proven from what -- but that is a massive oversimplification, to say the least.
Is Logic a Part of Philosophy?
Now that I've spent a whole lot of time not answering your question, what is the answer I would give? From my view, formal logic is so interweaved into modern analytic philosophy that any definition of philosophy which excluded it would be defective beyond recognition.
So, yes, formal logic is a part of philosophy. But also of mathematics. And linguistics. And computer science. (And probably some other fields I'm unaware of.) All of these branches make substantive contributions to the existing body of literature on formal logic but none can claim exclusive dominion over it.