Small Functions in Disks of $\C_p$

Small functions were defined in complex analysis and next in ultrametric analysis. Order of growth and type of growth were also defined in complex analysis and have a similar definition in ultrametric analysis. Here we compare these two notions in the same way, on a complete ultrametric algebraically closed field $\K$ of characteristic $0$ such as $\C_p$. Small functions with respect to an entire function $f$ were studied in several articles. Inside an ''open'' disk, small functions also exist. After a general study, here we examine how two analytic functions inside an open disk can share three small functions, ignoring multiplicity and we give sufficient conditions proving that these two functions are equal.