  Brown dwarfs and giant planets orbiting close to a host star are subjected to
significant irradiation that can modify the properties of their atmospheres. In
order to test the atmospheric models that are used to describe these systems,
it is necessary to obtain accurate observational estimates of their physical
properties (masses, radii, temperatures, albedos). Interacting compact binary
systems provide a natural laboratory for studying strongly irradiated
sub-stellar objects. As the mass-losing secondary in these systems makes a
critical, but poorly understood transition from the stellar to the sub-stellar
regime, it is also strongly irradiated by the compact accretor. In fact, the
internal and external energy fluxes are both expected to be comparable in these
objects, providing access to an unexplored irradiation regime. However, the
atmospheric properties of such donors have so far remained largely unknown.
Here, we report the direct spectroscopic detection and characterisation of an
irradiated sub-stellar donor in an accreting white dwarf binary system. Our
near-infrared observations allow us to determine a model-independent mass
estimate for the donor of $M_2=0.055\pm0.008M_{\odot}$ and an average spectral
type of ${\rm L1}\pm{\rm1}$, supporting both theoretical predictions and
model-dependent observational constraints. Our time-resolved data also allow us
to estimate the average irradiation-induced temperature difference between the
day and night sides on the sub-stellar donor, $\Delta{\rm T}\simeq57$~K, and
the maximum difference between the hottest and coolest parts of its surface, of
$\Delta{\rm T}_{max}\simeq200$~K. The observations are well described by a
simple geometric reprocessing model with a bolometric (Bond) albedo of
$A_B<0.54$ at 2-$\sigma$ confidence level, consistent with high reprocessing
efficiency, but poor lateral heat redistribution in the donor's atmosphere.
