A Semmes surface in the Heisenberg group is a closed set
S
S
that is upper Ahlfors-regular with codimension one and satisfies the following condition, referred to as Condition B. Every ball
B
(
x
,
r
)
B(x,r)
with
x
∈
S
x \in S
and
0
>
r
>
diam
S
0 > r > \operatorname {diam} S
contains two balls with radii comparable to
r
r
which are contained in different connected components of the complement of
S
S
. Analogous sets in Euclidean spaces were introduced by Semmes in the late 1980s. We prove that Semmes surfaces in the Heisenberg group are lower Ahlfors-regular with codimension one and have big pieces of intrinsic Lipschitz graphs. In particular, our result applies to the boundary of chord-arc domains and of reduced isoperimetric sets. The proof of the main result uses the concept of quantitative non-monotonicity developed by Cheeger, Kleiner, Naor, and Young. The approach also yields a new proof for the big pieces of Lipschitz graphs property of Semmes surfaces in Euclidean spaces.