We construct the example of the title.

We prove that if 0 < p < 1 then a normalized unconditional basis of a complemented subspace of $c_0\left(l_p\right)$ must be equivalent to a permutation of a subset of the canonical unit vector basis of $c_0\left(l_p\right)$. In particular, $c_0\left(l_p\right)$ has unique unconditional basis up to permutation. Bourgain, Casazza, Lindenstrauss, and Tzafriri have previously proved the same result for $c_0\left(l₁\right)$.

We prove that the quasi-Banach spaces ${\ell }_p\left(c₀\right)$ and ${\ell }_p\left(\ell ₂\right)$ (0 < p < 1) have a unique unconditional basis up to permutation. Bourgain, Casazza, Lindenstrauss and Tzafriri have previously proved that the same is true for the respective Banach envelopes $\ell ₁\left(c₀\right)$ and ℓ₁(ℓ₂). They used duality techniques which are not available in the non-locally convex case.

The aim of the present paper is to study the class of tvs which we define by ommiting the word increasing in the definition of *-suprabarrelled spaces. We prove that the product of Baire tvs is *-UBL and hence the class of *-UBL spaces is stricty larger than the class of Baire spaces.