Geometric group theory is mainly concerned with topological and geometric properties of groups, spaces on which they act etc., so the ideas employed in GGT are mainly algebraic/geometric/topological. Is there any subfield of GGT where methods from analysis find applications? I once heard that analytical tools, e. g. geodesic flows, are used in studying ends of groups, but that's all I know.
Analytic ideas enter into several parts of geometric group theory. Tom already mentioned amenability, so I'll skip that.
1) Complex analysis and the theory of quasiconformal mappings plays an important role in understanding the mapping class group, which is one of the most important groups studied by geometric group theorists. For information on this, see Farb and Margalit's forthcoming book "A primer on mapping class groups", available on either of their webpages.
2) The study of groups with property (T) ends up using quite a bit of analysis. See the book "Kazhdan's Property (T)" by de la Harpe, Bekka, and Valette for information on this.
3) Analysis (together with ideas from ergodic theory, which is of course quite analytic) plays an important role in proving various rigidity theorems. The most famous is the Mostow Rigidity Theorem, whose original proof uses lots of analysis : quasiconformal mapping in high dimensions, the fact that Lipschitz functions are differentiable almost everywhere, ergodic theory, etc.
I'll expand a little on Andy and Tom's very nice answers.
Property (T), amenability and rigidity theorems are not really separate topics. For a beautiful introduction, including some of the connections between these and other topics, see Lubotzky's book "Discrete Groups, Expanding Graphs, and Invariant Measures". Another great reference is Zimmer's book "Ergodic Theory and Semisimple Groups".
As for whether this sort of stuff counts as a "subfield" of geometric group theory, that probably depends on your definition of geometric group theory.
One point of contact between GGT and analysis is the study of amenability. See for instance the wikipedia article on amenable groups. As you can read there, there are many equivalent conditions for a (discrete) group to be amenable, some couched in terms of analytic concepts. On the other hand, if you have a look at the usual proof that the free group on two generators is not amenable, that has a definite flavour of geometric group theory about it.
(There is also the saga of whether Thompson's group $F$ is amenable. Danny Calegari has an excellent blog post on this. In short, the situation is this. For many years, the question of whether $F$ is amenable has been an open question. But in 2009, one paper appeared purporting to prove that $F$ is amenable, and another paper appeared purporting to prove that $F$ is not amenable. The one claiming the positive result uses a kind of analysis rarely seen in GGT, and so received a lot of attention. As far as I can tell, both authors have now admitted serious flaws in their papers, but neither has withdrawn it from the arXiv.)
The work of Yu, partly joint with Kasparov, connects geometric group theory to the geometry of Banach spaces: If a finitely generated group embeds coarsely (in the sense of Gromov) into a uniformly convex Banach space, then it satisfies the coarse Baum-Connes conjecture. In effect, this means you can work on certain problems in geometric group theory without knowing the definition of a group (e.g. by determining when a metric space with bounded geometry coarsely embeds into a uniformly convex Banach space).