$\ell^p$ is not isometric to $\ell^q$

Remark As is well-known, the Banach spaces $c_0$ and $\ell^p$ for $1\leq p<\infty$ are mutually non-isomorphic, as a consequence of Pitt's theorem. See Corollary 2.1.6 in Albiac-Kalton's book. It seems a bit easier to prove the weaker result saying that the $\ell^p$ spaces are mutually non-isometric.

Note that $\ell^1$ is not strictly convex, while Clarkson's inequalities show that $\ell^p$ is strictly convex for every $1<p<\infty$. So $\ell^1$ is not isometric to $\ell^p$ for any $1<p<\infty$. We could obtain the same conclusion using reflexivity or uniform convexity instead of strict convexity.

Now assume there exist $1<p,q<\infty$ such that $\ell^p$ and $\ell^q$ are isometric via some isometry $T$: $\|Tx\|_p=\|x\|_q$ for every $x\in \ell^q$. We need to show that $p=q$.

Sketch An elementary manipulation of Clarkson's inequalities (see here for a short proof by R.P. Boas) yields Claim 1 below. Claims 2,3,4 follow immediately. This allows us to rule out every case but $q=p$ and $q=p'$. So it only remains to check that $\ell^2$ is the only $\ell^p$ space which is isometric to its dual (i.e. Claim 5, essentially).

We will denote $p'$ the conjugate exponent associated with $p$, i.e. $\frac{1}{p}+\frac{1}{p'}=1 \iff p'=\frac{p}{p-1}$. Recall that the continuous dual $(\ell^p)'$ is isometric to $\ell^{p'}$ for every $1<p<\infty$.

Claim 1 If $p\geq 2$, then $p'\leq q\leq p$.

Proof For every $x,y \in \ell^q$, we have $$ \big\| \frac{x+y}{2} \big\|_q^p+\big\| \frac{x-y}{2} \big\|_q^p=\big\| \frac{Tx+Ty}{2} \big\|_p^p+\big\| \frac{Tx-Ty}{2} \big\|_p^p $$ $$ \leq \frac{\|Tx\|_p^p+\|Ty\|_p^p}{2}=\frac{\|x\|_q^p+\|y\|_q^p}{2} $$ where we used Clarkson's inequality for $p\geq 2$. In particular, for $x=(1,1,0,\ldots)$ and $y=(1,-1,0,\ldots)$, this yields $$2=1+1\leq \frac{2^\frac{p}{q}+2^\frac{p}{q}}{2}=2^\frac{p}{q}\quad\Rightarrow\quad q\leq p$$ While for $x=(2,0,\ldots)$ and $y=(0,2,0,\ldots)$, we get $$ 2\cdot 2^\frac{p}{q}=2^\frac{p}{q}+2^\frac{p}{q} \leq \frac{2^p+2^p}{2}=2^p \quad\Rightarrow\quad p'=\frac{p}{p-1}\leq q $$ Therefore we have $p'\leq q\leq p$. $\Box$

Claim 2 If $p\leq 2$, then $p\leq q\leq p'$.

Proof Taking the dual, $\ell^p\simeq \ell^q$ yields an isometric isomorphism $\ell^{p'}\simeq \ell^{q'}$. Since $p'\geq 2$, Claim 1 applied to $p'$ implies $(p')'\leq q'\leq p'\iff p\leq q\leq p'$. $\Box$

Claim 3 If $p$ and $q$ are both $\geq 2$ or both $\leq 2$, then $p=q$.

Proof Assume $p,q\geq 2$. Applying Claim 1 to $p$ yields $q\leq p$, while applying it to $q$ gives $p\leq q$. Hence $p=q$. The case $p,q \leq 2$ follows from Claim 2 in a similar manner. Or you can just take the duals and conclude that $p'=q'$ since they are both $\geq 2$. $\Box$

Claim 4 If $p\geq 2$ and $q\leq 2$, then $q=p'$.

Proof By Claim 1 for $p$, we have $p'\leq q$. By Claim 2 for $q$, we get $p\leq q' \iff q\leq p'$. Hence $q=p'$. $\Box$

Claim 5 If $\ell^p$ is isometric to $\ell^{p'}$, then $p=p'=2$.

Proof Still pondering what the easiest argument could be...

[Edit by the OP: see my answer below, which should prove Claim 5 and finish the problem off.]

Let me complete the accepted answer by showing that $\ell^p$ is not isometric to $\ell^q$ if $1<p<2$ and $2<q<\infty$. Fix $1<r<\infty$. I claim that

Claim 5' The second derivative $\frac{d^2}{dt^2}\|x+ty\|_{\ell^r}\Big|_{t=0}$ exists for every $x\in\ell^r\setminus\{0\}$ and every $y\in\ell^r$ if and only if $r\ge 2$.

Clearly, the claim implies the first assertion and thus finishes the problem.

Proof If $r<2$ then we take $x:=e_1$, $y:=e_2$ and we see that the second derivative $\frac{d^2}{dt^2}(1+|t|^r)^{1/r}\Big|_{t=0}$ does not exist. On the other hand, let $r\ge 2$: we claim that $$\begin{aligned}\frac{d}{dt}\|x+ty\|_{\ell^r}\Big|_{t=0}&=\sum_{n=1}^\infty x'_ny_n=:A(x,y), \\ \frac{d^2}{dt^2}\|x+ty\|_{\ell^r}\Big|_{t=0}&=(r-1)\sum_{n=1}^\infty x''_ny_n^2-\frac{(r-1)}{\|x\|}\Big(\sum_{n=1}^\infty x'_ny_n\Big)^2=:B(x,y),\end{aligned}$$ where $x'\in\ell^{r/(r-1)}$ and $x''\in\ell^{r/(r-2)}$ are defined by $$x'_n:=\frac{|x_n|^{r-2}x_n}{\|x\|_{\ell^r}^{r-1}},\quad x''_n:=\frac{|x_n|^{r-2}}{\|x\|_{\ell^r}^{r-1}}.$$ Notice that $x'$ and $x''$ depend continuously on $x\neq 0$ (which is easily seen by dominated convergence) and $\|x'\|_{\ell^{r/(r-1)}}=1$, $\|x''\|_{\ell^{r/(r-2)}}=\|x\|_{\ell^r}^{-1}$. So, by the generalized Holder's inequality, $A(x,y)$ and $B(x,y)$ are continuous and satisfy $|A(x,y)|\le\|y\|_{\ell^r}$, $|B(x,y)|\le 2(r-1)\|x\|_{\ell^r}^{-1}\|y\|_{\ell^r}^2$.

These formulas are a straightforward computation if $x,y\in c_c$ ($c_c$ being the space of sequences where only finitely many terms are nonzero). In order to check that they are true in general, we observe that for $x,y\in c_c$ $$\|x+ty\|_{\ell^r}=\|x\|_{\ell^r}+tA(x,y)+\int_0^t(t-s)B(x+sy,y)\,ds$$ (Taylor's expansion with remainder in integral form) and, by a density argument and dominated convergence, we see that this equation holds for all $x\in\ell^r\setminus\{0\}$ and all $y\in\ell^r$. Hence, $$\begin{aligned}\frac{d}{dt}\|x+ty\|_{\ell^r}&=A(x,y)+\int_0^t B(x+sy,y)\,ds, \\ \frac{d^2}{dt^2}\|x+ty\|_{\ell^r}\Big|_{t=0}&=B(x,y).\end{aligned}$$ This finishes the proof. $\square$