Is pressure extensive or intensive?
You are right that intensive properties are not additive like the properties which depend on the extent of the system. Pressure is indeed an intensive property because it is force divided by an extensive quantity (area). Now partial pressure is an interesting property. It is a hypothetical construct which cannot be directly or &quot;experimentally&quot; measured in a mixture of gases yet it can be calculated for individual gases in that mixture.
The basis of calculation of partial pressure is the number of particles in the container of each gas (hence an partial pressure originates from the number of particles of gas-A and gas-B).  Greater the number of particles of each gas in a container of fixed volume, greater is the calculated partial pressure of each component. The number of particles in a container determine the partial pressure which is an extensive property and hence partial pressures are additive.
Wikipedia explains the limitations of classifying quantities as extensive or intensive.
Consider reading the Journal of Chemical Education Article (free download from Google Scholar)
Redlich, &quot;Intensive and Extensive Properties&quot; J. Chem. Educ., 1970, 47 (2), 154–156. Redlich says that this classification is an incredible source of confusion in thermodynamics. I feel that he is not wrong.

The general validity of the division of physical properties into extensive and intensive kinds has been addressed in the course of science. Redlich noted that, although physical properties and especially thermodynamic properties are most conveniently defined as either intensive or extensive, these two categories are not all-inclusive and some well-defined physical properties conform to neither definition. Redlich also provides examples of mathematical functions that alter the strict additivity relationship for extensive systems, such as the square or square root of volume, which may occur in some contexts, albeit rarely used.

&gt; Other systems, for which standard definitions do not provide a simple
&gt; answer, are systems in which the subsystems interact when combined.
&gt; Redlich pointed out that the assignment of some properties as
&gt; intensive or extensive may depend on the way subsystems are arranged.
&gt; For example, if two identical galvanic cells are connected in
&gt; parallel, the voltage of the system is equal to the voltage of each
&gt; cell, while the electric charge transferred (or the electric current)
&gt; is extensive. However, if the same cells are connected in series, the
&gt; charge becomes intensive and the voltage extensive. The IUPAC
&gt; definitions do not consider such cases.
&gt; 
&gt; Some intensive properties do not apply at very small sizes. For
&gt; example, viscosity is a macroscopic quantity and is not relevant for
&gt; extremely small systems. Likewise, at a very small scale color is not
&gt; independent of size, as shown by quantum dots, whose color depends on
&gt; the size of the &quot;dot&quot;.