We are analyzing a real signal with the DTFT. Since we are using a limited number of samples it's like we are transforming a finite signal.
No. The DTFT takes an infinite discrete time signal as an input but the spectrum is continuous. I think you are confusing the DTFT and the DFT. They are different things.
As I remember, the FT of a finite signal has an infinite band spectrum.
So I suspect that the DTFT of a sampled signal (with a finite number of samples) always produce aliasing (since in the DTFT spectrum each replica is an infinite band). Is this correct?
Yes and no. Sampling a finite signal does with a finite sampling frequency will always induce some amount of aliasing, although in practice you can make it almost always "small enough" so it's acceptable for your application. That has nothing to do with DTFT (or DFT). Aliasing is caused by sampling, not during the analysis of the sampled data. IN other words, the aliasing happens regardless of whether you apply a DTFT or not.