  The nature of the progenitor systems and explosion mechanisms that give rise
to Type Ia supernovae (SNe Ia) are still debated. The interaction signature of
circumstellar material (CSM) being swept up by expanding ejecta can constrain
the type of system from which it was ejected. Most previous studies have
focused on finding CSM ejected shortly before the SN Ia explosion still
residing close to the explosion site, resulting in short delay times until the
interaction starts. We use a sample of 3627 SNe Ia from the Zwicky Transient
Facility discovered between 2018 and 2020 and search for interaction signatures
over 100 days after peak brightness. By binning the late-time light curve data
to push the detection limit as deep as possible, we identify potential
late-time rebrightening in 3 SNe Ia (SN 2018grt, SN 2019dlf, SN 2020tfc). The
late-time detections occur between 550 and 1450 d after peak brightness, have
mean absolute $r$-band magnitudes of -16.4 to -16.8 mag and last up to a few
hundred days, significantly brighter than the late-time CSM interaction
discovered in the prototype SN 2015cp. The late-time detections all occur
within 0.8 kpc of the host nucleus and are not easily explained by nuclear
activity, another transient at a similar sky position, or data quality issues.
This suggests environment or specific progenitor characteristics playing a role
in producing potential CSM signatures in these SNe Ia. By simulating the ZTF
survey we estimate that <0.5 per cent of normal SNe Ia display late-time strong
H $\alpha$-dominated CSM interaction. This is equivalent to an absolute rate of
$8_{-4}^{+20}$ to $54_{-26}^{+91}$ Gpc$^{-3}$ yr$^{-1}$ assuming a constant SN
Ia rate of $2.4\times10^{-5}$ Mpc$^{-3}$ yr$^{-1}$ for $z \leq 0.1$. Weaker
interaction signatures, more similar to the strength seen in SN 2015cp, could
be more common but are difficult to constrain with our survey depth.
