Let ξ_t, t∈[0,T], be a strong Markov process with values in a complete separable metric space (X,ρ) and with transition probability function P_s,t(x,dy), 0≤s≤t≤T, x∈X. For any h∈[0,T] and a>0, consider the function $$α(h,a)=\sup\{P_{s,t}(x,\{y:ρ(x,y)≥a\}):x∈X,0≤s≤t≤(s+h)∧T\}.$$ It is shown that a certain growth condition on α(h,a), as a↓0 and h stays fixed, implies the almost sure boundedness of the p-variation of ξ_t, where p depends on the rate of growth.