The results of an amplitude analysis of the charmless three-body decay
B
+
→
π
+
π
+
π
−
, in which
C
P
-violation effects are taken into account, are reported. The analysis is based on a data sample corresponding to an integrated luminosity of
3
 
 
fb
−
1
of
p
p
collisions recorded with the LHCb detector. The most challenging aspect of the analysis is the description of the behavior of the
π
+
π
−
 
 
S
-wave contribution, which is achieved by using three complementary approaches based on the isobar model, the
K
-matrix formalism, and a quasi-model-independent procedure. Additional resonant contributions for all three methods are described using a common isobar model, and include the
ρ
(
770
)
0
,
ω
(
782
)
and
ρ
(
1450
)
0
resonances in the
π
+
π
−

P
-wave, the
f
2
(
1270
)
resonance in the
π
+
π
−

D
-wave, and the
ρ
3
(
1690
)
0
resonance in the
π
+
π
−

F
-wave. Significant
C
P
-violation effects are observed in both
S
- and
D
-waves, as well as in the interference between the
S
- and
P
-waves. The results from all three approaches agree and provide new insight into the dynamics and the origin of
C
P
-violation effects in
B
+
→
π
+
π
+
π
−
decays.