The cape can provide a benefit more than one time
The feature in question states:
The muddy cape allows him to ignore the first 10 points of fire damage from any attack or source.
Note that there are several other features in the rules that use the words "any attack":
Cavalier Fighter's Unwavering Mask feature:
[...] While it is within 5 feet of you, a creature marked by you has disadvantage on any attack roll that doesn't target you [...]
Ancestral Guardian Barbarian's Ancestral Protectors feature:
[...] Until the start of your next turn, that target has disadvantage on any attack roll that isn't against you [...]
Hexblade Warlock's Hexblade's Curse feature:
[...] Any attack roll you make against the cursed target is a critical hit on a roll of 19 or 20 on the d20 [...]
All of these features can apply more than one time. Compare them to the Divination Wizard's foretelling rolls from Portent:
[...] You can replace any attack roll, saving throw, or ability check made by you or a creature that you can see with one of these foretelling rolls. [...]
Each foretelling roll can be used only once. When you finish a long rest, you lose any unused foretelling rolls.
This feature specifically states that though it can happen to any attack roll it can only happen once. The muddy cape has no such restriction and thus, like the other features above, it can provide its benefit multiple times.