TITLE: Product-one sets in non-commutative groups
QUESTION [7 upvotes]: A nonempty subset $D$ of a group $G$ is called
$\bullet$ decomposable if $D\subseteq DD$, that is every element $x\in D$ is can be written as the product $x=yz$ of some elements $y,z\in D$;
$\bullet$ product-one if there exists $n\in\mathbb N$ and pairwise distinct elements $x_1,\dots,x_n\in D$ such that $x_1\cdots x_n=1$.

Problem 1. Let $D$ be a finite decomposable subset of a group. Is $D$ product-one?

Remark 1. For commutative groups this problem was posed by Gjergji Zaimi and solved affirmatively by Lev, Nagy, and Pach.
Remarks 2. For some non-commutative groups like generalized dihedral groups the answer to Problem is also affirmative, see my partial answer below. This partial answer suggests the following

Problem 2. Let $G$ be a group containing an Abelian subgroup of index 2. Is every finite decomposable set in $G$ product-one?

REPLY [3 votes]: GAP shows that the groups SmallGroup(27,3), SmallGroup(27,4), SmallGroup(36,11), SmallGroup(39,1) SmallGroup(48,3) do contain many 5-element decomposable sets, which are not product-one. So, the lower bound 5 for the smallest cardinality of a counterexample, obtained by @YCor in his answer, is the best possible.
Below I write down 5-element decomposable non-product-one sets found by GAP in the groups
SmallGroup(27,3): [ f1, f2, f1 * f2, f1^2 * f2, f1 * f2^2 ]
SmallGroup(27,4): [ f1, f2, f1 * f2 * f3, f1^2 * f2 * f3, f2^2 * f3^2 ]
SmallGroup(36,11): [ f1, f2 * f3, f1^2 * f3, f1 * f2^2 * f3, f1^2 * f2^2 * f4 ]
SmallGroup(39,1):  [ f1, f2, f1 * f2, f1^2 * f2, f2^4 ]
SmallGroup(48,3): [ f1, f2, f1 * f2, f2 * f3, f1^2 * f2 ]
These 5 groups are the only groups of order $\le 50$ that contain decomposable non-product-one sets.