@skd Thanks, that article proves that $\Sigma^{\infty} (BS^{1})^{\wedge}_p$ is not $p$-complete, what made me think that $BG^{\wedge p}$ in the stable homotopy context means $(\Sigma^{\infty} BG)^{\wedge p}$, although it seems counterintuitive to me.

@user43326 Thank you!, I checked the characterization of $E_{\ast}$-localization according to Bousfield's paper, and indeed this property follows from that.

@user43326 No, I meant this property is well known on topological spaces, but besides the common name 'p-completion', its definition on spectra looks quite different.

@user43326 Actually I had the suspicion that (under some restrictions) $(H\mathbb{F}_p)_\ast(f)$ is an isomorphism if and only if $f^{\wedge p}$ is a stable homotopy equivalence.

@LennartMeier Do you mean a map $f$ with $H_\ast(f,\mathbb{F}_p)=0$ but $f^{\wedge p}\not\simeq\ast$?

I also suspected that some conditions on X were necessary, but these notes left me puzzled (see page 4), thank you!.

@LSpice thank you.