As mentioned by Scholze in this lecture (around 40:00), given a directed system of maps of uniform Tate-Huber pairs $(R_i,R_i^+) \to (S,S^+)$, the map $\varinjlim_i R_i \to S$ is continuous for the $\varpi$-adic topology on $\varinjlim_i R_i^+$, where $\varpi$ is the image of a pseudouniformizer of some $R_i$. Here we need to use the fact that $S$ is uniform, so $S^+$ is a ring of definition and it has the $\varpi$-adic topology.

So filtered inverse limits should exist in the category of (sous)perfectoid spaces since (sous)perfectoid rings are stably uniform. Unfortunately I do not know of a reference that explicitly proves this, but it does seem to be used implicitly in Etale cohomolgy of diamonds.

As mentioned by Scholze in this lecture (around 40:00; uniform Tate rings are called "spectral rings" in the lecture), given a directed system of maps of uniform Tate-Huber pairs $(R_i,R_i^+) \to (S,S^+)$, the map $\varinjlim_i R_i \to S$ is continuous for the $\varpi$-adic topology on $\varinjlim_i R_i^+$, where $\varpi$ is the image of a pseudouniformizer of some $R_i$. Here we need to use the fact that $S$ is uniform, so $S^+$ is a ring of definition and it has the $\varpi$-adic topology.

So filtered inverse limits should exist in the category of (sous)perfectoid spaces since (sous)perfectoid rings are stably uniform. Unfortunately I do not know of a reference that explicitly proves this, but it does seem to be used implicitly in Etale cohomolgy of diamonds.

Scholze mentions in this lecture (around 40:00) that there is a canonical way of defining a topology on a limit of uniform Tate-Huber pairs (uniform Tate rings are called "spectral rings" in the lecture): give the ring of integral elements $R^+$ the $\varpi$-adic topology, where $\varpi$ is a pseudouniformizer.

So inverse limits should exist in the category of perfectoid spaces (as well as in the category of sousperfectoid spaces) since (sous)perfectoid rings are stably uniform. Unfortunately I do not know of a reference that explicitly proves this, but it does seem to be used implicitly in Etale cohomolgy of diamonds.

As mentioned by Scholze in this lecture (around 40:00), given a directed system of maps of uniform Tate-Huber pairs $(R_i,R_i^+) \to (S,S^+)$, the map $\varinjlim_i R_i \to S$ is continuous for the $\varpi$-adic topology on $\varinjlim_i R_i^+$, where $\varpi$ is the image of a pseudouniformizer of some $R_i$. Here we need to use the fact that $S$ is uniform, so $S^+$ is a ring of definition and it has the $\varpi$-adic topology.

So filtered inverse limits should exist in the category of (sous)perfectoid spaces since (sous)perfectoid rings are stably uniform. Unfortunately I do not know of a reference that explicitly proves this, but it does seem to be used implicitly in Etale cohomolgy of diamonds.