  Nonstandard self-interactions can alter the evolution of cosmological
neutrinos, mainly by damping free streaming, which should leave traces in
cosmological observables. Although overall effects are opposite to those
produced by neutrino mass and a larger $N_{\rm eff}$, they cannot be totally
canceled by these last. We harness cosmological data that includes Cosmic
Microwave Background from Plank 2018, BAO measurements, local $H_0$,
Ly-$\alpha$ and SNIa, to constrain massive neutrino self-interactions with a
very light scalar mediator. We find that the effective coupling constant, at
the 95\% C.L., should be $g_{\rm eff}< 1.94 \times 10^{-7}$ for only Planck
2018 data and $1.97\times10^{-7}$ when Planck + BAO are considered. This bound
relaxes to $2.27\times 10^{-7}$ ($2.3\times 10^{-7}$) for $H_0$
($H_0$+SNe+Ly-$\alpha$) data. Using the Planck + BAO dataset, the $H_0$ tension
lowers from 4.3$\sigma$ (for $\Lambda$CDM) to 3.2$\sigma$. The Akaike
Information Criterion penalizes the self-interacting model due to its larger
parameter space for Plank or Planck + BAO data, but favors the interacting
model when we use local $H_0$ measurements. A somewhat larger value for $H_0$
is preferred when we include the whole data pool, which comes accompanied with
a larger value of $N_{\rm eff}$ and a more constricted bound on $\Sigma m_\nu$.
