Using multiple rewrite rules?

You must add [L] flag to stop reading rules when one match :

<IfModule mod_rewrite.c>
     RewriteEngine on

     RewriteCond %{REQUEST_FILENAME} !-f
     RewriteCond %{REQUEST_FILENAME} !-d

     RewriteRule ^p$ index.php?p= [L]
     RewriteRule ^(.*)$ index.php?s=$1 [L]
</IfModule>

Reminder of RewriteRule flags :

• L = Last. Stop processing RewriteRules once this one matches. Order counts!
• C = Chain. Continue processing the next RewriteRule. If this rule doesn't match, then the next rule won't be executed. More on this later.
• E = Set environmental variable. Apache has various environmental variables that can affect web-server behavior.
• F = Forbidden. Returns a 403-Forbidden error if this rule matches.
• G = Gone. Returns a 410-Gone error if this rule matches.
• H = Handler. Forces the request to be handled as if it were the specified MIME-type.
• N = Next. Forces the rule to start over again and re-match. BE CAREFUL! Loops can result.
• NC = No case. Allows [jpg] to match both jpg and JPG.
• NE = No escape. Prevents the rewriting of special characters (. ? # & etc) into their hex-code equivalents.
• NS = No subrequests. If you're using server-side-includes, this will prevent matches to the included files.
• P = Proxy. Forces the rule to be handled by mod_proxy. Transparently provide content from other servers, because your web-server fetches it and re-serves it. This is a dangerous flag, as a poorly written one will turn your web-server into an open-proxy and That is Bad.
• PT = Pass Through. Take into account Alias statements in RewriteRule matching.
• QSA = QSAppend. When the original string contains a query (http://example.com/thing?asp=foo) append the original query string to the rewritten string. Normally it would be discarded. Important for dynamic content.
• R = Redirect. Provide an HTTP redirect to the specified URL. Can also provide exact redirect code [R=303]. Very similar to RedirectMatch, which is faster and should be used when possible.
• S = Skip. Skip this rule.
• T = Type. Specify the mime-type of the returned content. Very similar to the AddType directive.

Extract from this (very) complete post : https://serverfault.com/questions/214512/everything-you-ever-wanted-to-know-about-mod-rewrite-rules-but-were-afraid-to-as

You should know first that the rewrite conditions only affect the following rewrite rule, an you added your new rule between the rewrite conditions and the rewrite rule, that means they will now affect your new rule only and not the old one (what you have in you code is that the rewrite rules are only executed if the targeted url is not a file or a directory), so if you want your old rule to be still affected by the rewrite condition, you will have to add your new rule before the rewrite conditions.

For your issue, I think zessx has answered enough (It is fixed by adding the [L] flag).

In the end you should have something like this :

<IfModule mod_rewrite.c>
     RewriteEngine on

     RewriteRule ^p$ index.php?p= [L]

     RewriteCond %{REQUEST_FILENAME} !-f
     RewriteCond %{REQUEST_FILENAME} !-d
     RewriteRule ^(.*)$ index.php?s=$1 [L]
</IfModule>