  Determining the habitability and interpreting atmospheric spectra of
exoplanets requires understanding their atmospheric physics and chemistry. We
use a 3-D Coupled Climate-Chemistry Model, the Met Office Unified Model with
the UK Chemistry and Aerosols framework, to study the emergence of lightning
and its chemical impact on tidally-locked Earth-like exoplanets. We simulate
the atmosphere of Proxima Centauri b orbiting in the Habitable Zone of its
M-dwarf star, but the results apply to similar M-dwarf orbiting planets. Our
chemical network includes the Chapman ozone reactions and hydrogen oxide
(HO$_{\mathrm{x}}$=H+OH+HO$_2$) and nitrogen oxide
(NO$_{\mathrm{x}}$=NO+NO$_2$) catalytic cycles. We find that photochemistry
driven by stellar radiation (177-850 nm) supports a global ozone layer between
20-50 km. We parameterise lightning flashes as a function of cloud-top height
and the resulting production of nitric oxide (NO) from the thermal
decomposition of N$_2$ and O$_2$. Rapid dayside convection over and around the
substellar point results in lightning flash rates of up to 0.16 flashes
km$^{-2}$yr$^{-1}$, enriching the dayside atmosphere below altitudes of 20 km
in NO$_{\mathrm{x}}$. Changes in dayside ozone are determined mainly by UV
irradiance and the HO$_{\mathrm{x}}$ catalytic cycle. ~45% of the planetary
dayside surface remains at habitable temperatures (T$_{\mathrm{surf}}$>273.15
K) and the ozone layer reduces surface UV radiation levels to 15%.
Dayside-nightside thermal gradients result in strong winds that subsequently
advect NO$_{\mathrm{x}}$ towards the nightside, where the absence of
photochemistry allows NO$_{\mathrm{x}}$ chemistry to involve reservoir species.
Our study also emphasizes the need for accurate UV stellar spectra to
understand the atmospheric chemistry of exoplanets.
