Do the interior lights under the dash work? Presume you've checked obvious things like interior light not going off when the door shuts due to faulty door switch? The old single filament globes used in the interior lights can be pretty dim and mightn't be obvious they're on in daylight at least.
Another possibility is that the dash lights are on as a result of turning the headlight switch all the way anti clockwise. This will also bring on the interior lights even with door shut but if the latter don't work, globes burnt out or missing, you'd be even less likely to notice your dash lights are on in daylight.
I flattened my battery by leaving the dash lights and therefore the interior lights on while checking the dash lights in daylight, just forgot to turn them off and never noticed the interior lights stayed on when I shut the door.
I don't have any expertise, but I gather a lot of electrical problems are caused by earthing issues. Might be worth checking both the negative battery to engine block connection and the other earth from rear of engine block or head to the firewall.
