Okay, first of all, apologies for my monologue here, but perhaps it will be helpful for some beginners.
I have now been able to solve my problem. I activated LISSAGE=βOUIβ (according to the translation, this means smoothing). That did the trick.
Here is the complete new COMM file again:
DEBUT(LANG='EN')
Mesh_Plate = LIRE_MAILLAGE(identifier='0:1',
UNITE=20)
Mesh_Stamp = LIRE_MAILLAGE(identifier='1:1',
UNITE=2)
Mesh = ASSE_MAILLAGE(identifier='2:1',
MAILLAGE_1=Mesh_Plate,
MAILLAGE_2=Mesh_Stamp,
OPERATION='SUPERPOSE')
Model = AFFE_MODELE(identifier='3:1',
AFFE=(_F(GROUP_MA=('Plate_Solid', ),
MODELISATION='3D',
PHENOMENE='MECANIQUE'),
_F(GROUP_MA=('Stamp_Solid', ),
MODELISATION='3D',
PHENOMENE='MECANIQUE')),
MAILLAGE=Mesh)
Material = DEFI_MATERIAU(identifier='4:1',
ELAS=_F(E=200000.0,
NU=0.3))
Section = AFFE_MATERIAU(identifier='5:1',
AFFE=_F(MATER=(Material, ),
TOUT='OUI'),
MODELE=Model)
Func = DEFI_FONCTION(identifier='6:1',
NOM_PARA='INST',
VALE=(0.0, 0.0, 1.0, 1.0))
List = DEFI_LIST_REEL(identifier='7:1',
DEBUT=0.0,
INTERVALLE=_F(JUSQU_A=1.0,
NOMBRE=10))
Fixture = AFFE_CHAR_MECA(identifier='8:1',
DDL_IMPO=_F(DX=0.0,
DY=0.0,
DZ=0.0,
GROUP_MA=('Fixture_Face', )),
MODELE=Model)
Load = AFFE_CHAR_MECA(identifier='9:1',
DDL_IMPO=_F(DZ=-2.0,
GROUP_MA=('Stamp_Solid', )),
MODELE=Model)
Contact = DEFI_CONTACT(identifier='10:1',
LISSAGE='OUI',
MODELE=Model,
ZONE=_F(GROUP_MA_ESCL=('Stamp_Refinement', ),
GROUP_MA_MAIT=('Plate_Refinement', )))
Stabilization = AFFE_CHAR_MECA(identifier='11:1',
DDL_IMPO=_F(DX=0.0,
DY=0.0,
GROUP_MA=('Stamp_Solid', )),
MODELE=Model)
Analysis = STAT_NON_LINE(identifier='12:1',
CHAM_MATER=Section,
COMPORTEMENT=_F(RELATION='ELAS'),
CONTACT=Contact,
EXCIT=(_F(CHARGE=Fixture),
_F(CHARGE=Load,
FONC_MULT=Func),
_F(CHARGE=Stabilization)),
INCREMENT=_F(LIST_INST=List),
MODELE=Model)
Analysis = CALC_CHAMP(identifier='13:1',
reuse=Analysis,
CRITERES=('SIEQ_ELGA', 'SIEQ_ELNO', 'SIEQ_NOEU'),
RESULTAT=Analysis)
IMPR_RESU(identifier='14:1',
FORMAT='MED',
RESU=_F(RESULTAT=Analysis),
UNITE=3)
FIN()

When I increased the displacement and the steps on a trial basis, problems arose again. This is due to the number of iterations required. So I set ITER_GEOM_MAXI to 50, and then a depth of 5 mm and 100 steps worked.
I would be grateful if a professional could tell me whether this analysis is OK, or whether I have made any mistakes or could improve anything.
Otherwise, thank you very much for your help π