We prove the existence of a polynomial of degree
d
d
defined on a closed subspace that cannot be extended to the Banach space
E
E
(in particular, the existence of a nonextendible polynomial) in the following cases: (1)
d
≥
2
d\geq 2
and
E
E
does not have type
p
p
for some
1
>
p
>
2
1>p>2
; (2) the space
ℓ
k
\ell _k
,
k
∈
N
k\in \mathbb {N}
,
2
>
k
≤
d
2>k\leq d
, is finitely representable in
E
E
. In each of these cases we prove, equivalently, the existence of a closed subspace
F
⊂
E
F\subset E
such that the subspace
⊗
^
s
,
π
d
F
\hat {\otimes }^{d}_{s,\pi }{F}
is not closed in
⊗
^
s
,
π
d
E
\hat {\otimes }^{d}_{s,\pi }{E}
.